Answers
a) The initial height is 5 feet.
b) The maximum height is 17 feet.
c) The horizontal displacement is 26.3 feet
How high is the ball when it left the childs hands?The height of the ball is modeled by the given quadratic equation:
y = (-1/12)x² + 2x + 5
The ball leaves the child's hands at the beginning, so to get the height at that point we just need to evaluate this in x = 0, we will get:
y = (-1/12)*0² + 2*0 + 5
y = 5
The height is 5 feet.
b) To get the maximum height we need to find the vertex, in this case the vertex is at:
x = -(2/2)*-12 = 12
Then the maximum height is:
y = (-1/12)*12² + 2*12 + 5 = 17 feet.
Lastly, the ball will hith the ground when y = 0, then we need to solve:
(-1/12)*x² + 2x + 5 = 0
x² -12*2x - 12*5 = 0
x² -24x - 60 = 0
The quadratic formula gives:
[tex]x = \frac{24 \pm \sqrt{(-24)^2 -4*1*-60} }{2}[/tex]
The positive solution gives:
x = 26.3
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Related Questions
The figure below displays the SAT scores of three students, but each chart looks different. The two charts have the same data, but the difference seems larger for the graph on the left. Why?
Answers
Answering the presented question, we may conclude that This greater expressions scale makes it easier to see the variations between the ratings of the three college students in a extra correct and informative way.
what is expression ?In mathematics, an expression is a collection of integers, variables, and complex mathematical (such as arithmetic, subtraction, multiplication, division, multiplications, and so on) that describes a quantity or value. Phrases can be simple, such as "3 + 4," or complicated, such as They may also contain functions like "sin(x)" or "log(y)". Expressions can be evaluated by swapping the variables with their values and performing the arithmetic operations in the order specified. If x = 2, for example, the formula "3x + 5" equals 3(2) + 5 = 11. Expressions are commonly used in mathematics to describe real-world situations, construct equations, and simplify complicated mathematical topics.
The difference in look between the two charts is due to the choice of the scales on the x and y-axes. In the left chart, the y-axis starts offevolved at 800 and has a range of only 200 points, whilst the x-axis starts offevolved at 1300 and has a vary of 200 points. This compressed scale makes the variations between the ratings of the three students appear larger than they absolutely are.
On the other hand, the proper chart has a y-axis that starts at zero and has a vary of 800 points, whilst the x-axis begins at 1200 and has a range of 800 points. This greater expanded scale makes it easier to see the variations between the ratings of the three college students in a extra correct and informative way.
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In the inequality, x (< with line underneath) 8, represents the number of books on a shelf.
How do you know if 8 is a possible value of x?
Answers
Answer:
The numbers 8 and lower are possible values of x.
Step-by-step explanation:
The inequality [tex]x\leq 8[/tex] means x is less than or equal to 8. Therefore, 8 is a possible value of x.
54 students, some study History and Government. n(G)=3x n(U)=54 n(H)=2x n(HnG)=x
i. How many students study both History and Government?
ii. How many students study only one subject?
Answers
Therefore, the number of students who study only one subject is n(H-only) + n(G-only) = 14 + 28 = 42.
What do you mean by set?In mathematics, a set is a collection of distinct objects, which are called its elements. These objects can be anything, such as numbers, letters, or even other sets. Sets are usually denoted by capital letters and the elements of a set are listed within braces, separated by.
Given by the question.
i. n (H ∩ G) can be found using the formula:
n(H ∩ G) = n(H) + n(G) - n(H U G)
where n (H U G) represents the number of students who study either History or Government or both.
We know that n(H) = 2x and n(G) = 3x. Also, n(U) = 54, which means the total number of students is 54. Therefore, we can write:
n (H U G) = n(H) + n(G) - n (H ∩ G)
54 = 2x + 3x - x
54 = 4x
x = 13.5
Since x must be a whole number, we can round it up to 14. Therefore, n (H ∩ G) = x = 14.
ii. The number of students who study only one subject can be found by subtracting n (H ∩ G) from n(H) and n(G) respectively, and then adding the number of students who study neither History nor Government.
n(H-only) = n(H) - n (H ∩ G) = 2x - x = x = 14
n(G-only) = n(G) - n (H ∩ G) = 3x - x = 2x = 28
n(Neither) = n(U) - n (H U G) = 54 - 14 = 40
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3. Each sample of water from a river has a 10% chance of contamination by a particular heavy metal. Find the probability that in 18 independent samples taken from the same river, only two samples were contaminated. [3 marks]
Answers
The probability that, out of 18 independent samples received from one river, just two were contaminated is 0.8438.
Explain about the independent samples?Randomly chosen samples are known as independent samples since their results are independent of other observations' values. The premise that sampling are independent underlies many statistical analysis.When each trial possesses the same probability of achieving a given value, the number of trials or observations is represented using the binomial distribution.In the following 18 samples to be evaluated,
Let X = the number of samples that now the pollutant is present in.
Thus, with p = 0.10 and n = 18, X is a binomial random variable.
Using the binomial theorem:
[tex](^{n} _{r} ) p^{x} q^{n-x}[/tex]
p = 0.10
q = 1 - 0.10 = 0.9
n = 18
The likelihood that only two samples out of 18 obtained in different ways from the same river were polluted
P(x = 2) = [tex](^{18} _{2} ) (0.1)^{2} (0.9)^{18-2}[/tex]
= [tex](^{18} _{2} ) (0.1)^{2} (0.9)^{16}[/tex]
= 153 x 0.01 x 0.1853
= 0.8438
Thus, the probability that, out of 18 separate samples received from one river, just two were contaminated is 0.8438.
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Given that (1, 2, 3] System{1, 4, 7,6] for a system known to be LTI, compute the system's impulse response h[n] without using z-transforms.
Answers
Given that (1, 2, 3] System{1, 4, 7,6] for a system known to be LTI, the impulse response of the system: h[n] = (1/2)*δ[n] + δ[n-1] + (3/2)*δ[n-2]
To compute the impulse response h[n] of a linear time-invariant (LTI) system given its input-output relationship, we can use the convolution sum:
y[n] = x[n] * h[n]
y[n] = (1/2)*(x[n] + 2x[n-1] + 3x[n-2])
y[n] = (1/2)*(δ[n] + 2δ[n-1] + 3δ[n-2])
y[n] = (1/2)*δ[n] + δ[n-1] + (3/2)*δ[n-2]
Thus, the impulse response of the system is:
h[n] = (1/2)*δ[n] + δ[n-1] + (3/2)*δ[n-2],where δ[n] is the impulse signal.
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b is the set of odd positive integers less than 11.
(a)list all the elements of b in set notation.
b)state whether each of the followning statements is true or false.
(i)1∈b
Answers
Answer:
Step-by-step explanation:
True. 1 is an odd positive integer less than 11 and is an element of set B.
Write (28)to the power of 3 as a power of 2
Answers
The expression (2^8) to the power of 3 as a power of 2 is 4096^2
Rewritting the expression as a power of 2Given the following expression
(2^8) to the power of 3
To write (2^8) to the power of 3 as a power of 2, we can use the rule of exponents that says:
(a^b)^c = (a^c)^b
Applying this rule to (2^8)^3, we get:
(2^8)^3 = (2^3)^8
Simplifying the expression, we get:
(2^8)^3 = (8)^8
So, we have
(2^8)^3 = (8^4)^2
Simplify
(2^8)^3 = 4096^2
Therefore, (2^8)^3 can be written as 4096^2.
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Help pleaseeee!!
On January 1, 2014, the federal minimum wage was $7.25 per hour. Which graph has a slope that best represents this rate?
Answers
The horizontal line at $7.25 on the y-axis of the graph is the one with a slope that most accurately depicts the federal minimum wage of $7.25 per hour as of January 1, 2014.
Which federal minimum wage was the highest?Although it varies from state to state, the federally mandated minimum wage in the United States is $7.25 per hour. The District of Columbia had the highest minimum wage in the US as of January 1, 2023, at 16.50 dollars per hour.
How are minimum wages determined?The variable dearness allowance (VDA) component, which takes into account inflationary trends, such as an increase or fall in the Consumer Price Index (CPI), and, if applicable, the housing rent, are included in the computation of the monthly minimum salary.
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Explain why it is likely that the distributions of the
following variables will be normal:
a) the volume of soft drinks in cans
b) the diameter of bolts immediately after manufacture.l
Answers
After answering the provided question, we can conclude that This expression process frequently results in bolt diameters that are normally distributed, with minor random variations around the target diameter.
what is expression ?In mathematics, an expression is a collection of symbols, digits, and companies that portray a statistical correlation or formula. An expression can be a single number, a mutable, or a combination of both of them. Addition, subtraction, proliferation, division, and exponentiation are examples of mathematical operators. Expressions are used extensively in mathematics, including arithmetic, calculus, and geometry. They are used in mathematical formula representation, equation solution, and mathematical relationship simplification.
a) Because of the central limit theorem, the volume of soft drinks in cans is likely to follow a normal distribution. Because soft drink manufacturers typically produce a large number of cans filled with the same volume of liquid, the sample size is large.
b) The diameter of bolts immediately after manufacture is also likely to be distributed normally. Bolt manufacturing typically involves a large number of measurements and adjustments to ensure that the bolts are manufactured to exact specifications. This process frequently results in bolt diameters that are normally distributed, with minor random variations around the target diameter.
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This is accounting. Please answer
Answers
All answers are mentioned below.
Describe balance sheet?A balance sheet is a financial statement that provides a snapshot of a company's financial position at a specific point in time. It presents the company's assets, liabilities, and equity, and shows how these are financed. The balance sheet follows the accounting equation, which states that the total assets must equal the sum of liabilities and equity.
(a) Trading, Profit & Loss and Profit & Loss Appropriation Account for the year ended December 31, 1996
Trading Account
$
Sales (51,550 - 5,550 + 5,385) 51,385
Less: Cost of Goods Sold
Opening Stock 5,550
Add: Purchases 29,525
35,075
Less: Closing Stock 5,385
Cost of Goods Sold 29,690
Gross Profit 21,695
Profit & Loss Account
$
Gross Profit 21,695
Less: Expenses
Carriage Inward 100
Discount Allowed 1,000
Wages & Salaries 17,650
Motor Vehicle Repairs 5,850
Telephones 350
Rent & Rates 1,800
Advertising 2,750
Carriage Outward 1,000
Bank Charges 550
Depreciation
Land & Building (30,000/50 years) 600
Motor Vehicle (20% x 20,000) 4,000
34,550
Net Loss (12,855)
Profit & Loss Appropriation Account
$
Net Loss (12,855)
Add: Interest on Capital (5% x $90,000) 4,500
(8,355)
To be shared equally between Knox and Cox
Knox (4,178)
Cox (4,178)
Additional Salary for Cox (5,000)
Carried Forward (13,356)
(b) Partners' Current Account
Knox Cox
$ $
To Opening Balance 8,185 4,400
By Share of Net Loss (4,178) (4,178)
By Interest on Capital 2,500 2,000
By Drawings (1,200) (500)
By Salary 0 (5,000)
To Closing Balance 5,307 (3,278)
(c) Balance Sheet as at December 31, 1996
Fixed Assets
Land & Building (100,000 - 30,000) 70,000
Motor Vehicles (20,000 - 4,000) 16,000
86,000
Current Assets
Stock 5,385
Debtors 7,250
Bank 3,200
Cash 510
16,345
Current Liabilities
Creditors (7,650)
Net Current Assets 8,695
94,695
Less: Partners' Capital and Current Accounts
Knox Capital (50,000 + 5,307) 55,307
Cox Capital (40,000 - 3,278) 36,722
92,029
2,666
Notes:
Depreciation of Land and Building: 30,000/50 years = 600 per year
Depreciation of Motor Vehicles: 20% x 20,000 = 4,000
Interest on Capital: Knox - 5% x 50,000 = 2,500, Cox - 5% x 40,000 = 2,000
Interest on Drawings: Knox - 10% x 1,200 = 120, Cox - 10% x 500 = 50
Additional Salary for Cox is treated as an expense and charged to the Profit & Loss Account.
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(a) Trading, Profit & Loss and Profit for the year ended December 31, 1996 is mentioned below. (b) Knox Cox, Opening Balance 8,185 4,400, respectively.(c) Interest on Drawings: Knox - 10% x 1,200 = 120, Cox - 10% x 500 = 50
Describe balance sheet?A balance sheet is a financial statement that provides a snapshot of a company's financial position at a specific point in time. It presents the company's assets, liabilities, and equity, and shows how these are financed. The balance sheet follows the accounting equation, which states that the total assets must equal the sum of liabilities and equity.
(a) Trading, Profit & Loss and Profit & Loss Appropriation Account for the year ended December 31, 1996
Trading Account
$
Sales (51,550 - 5,550 + 5,385) 51,385
Less: Cost of Goods Sold
Opening Stock 5,550
Add: Purchases 29,525
35,075
Less: Closing Stock 5,385
Cost of Goods Sold 29,690
Gross Profit 21,695
Profit & Loss Account
$
Gross Profit 21,695
Less: Expenses
Carriage Inward 100
Discount Allowed 1,000
Wages & Salaries 17,650
Motor Vehicle Repairs 5,850
Telephones 350
Rent & Rates 1,800
Advertising 2,750
Carriage Outward 1,000
Bank Charges 550
Depreciation
Land & Building (30,000/50 years) 600
Motor Vehicle (20% x 20,000) 4,000
34,550
Net Loss (12,855)
Profit & Loss Appropriation Account
Net Loss (12,855)
Add: Interest on Capital (5% x $90,000) 4,500
(8,355)
To be shared equally between Knox and Cox
Knox (4,178)
Cox (4,178)
Additional Salary for Cox (5,000)
Carried Forward (13,356)
(b) Partners' Current Account
Knox Cox
$ $
To Opening Balance 8,185 4,400
By Share of Net Loss (4,178) (4,178)
By Interest on Capital 2,500 2,000
By Drawings (1,200) (500)
By Salary 0 (5,000)
To Closing Balance 5,307 (3,278)
(c) Balance Sheet as at December 31, 1996
Fixed Assets
Land & Building (100,000 - 30,000) 70,000
Motor Vehicles (20,000 - 4,000) 16,000
86,000
Current Assets
Stock 5,385
Debtors 7,250
Bank 3,200
Cash 510
16,345
Current Liabilities
Creditors (7,650)
Net Current Assets 8,695
94,695
Less: Partners' Capital and Current Accounts
Knox Capital (50,000 + 5,307) 55,307
Cox Capital (40,000 - 3,278) 36,722
92,029
2,666
Notes:
Depreciation of Land and Building: 30,000/50 years = 600 per year
Depreciation of Motor Vehicles: 20% x 20,000 = 4,000
Interest on Capital: Knox - 5% x 50,000 = 2,500, Cox - 5% x 40,000 = 2,000
Interest on Drawings: Knox - 10% x 1,200 = 120, Cox - 10% x 500 = 50
Additional Salary for Cox is treated as an expense and charged to the Profit & Loss Account.
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In May 2022, Reginald graduated from the Naval Academy with a degree in aeronautical engineering and was assigned to Pensacola, Florida as a permanent duty station. In his move to Pensacola, Reginald incurred the following costs: $450 in gasoline. $250 for renting a truck from UPAYME rentals. $100 for a tow trailer for his car. $90 in food. $35 in double espressos from Starbucks. $300 for motel lodging on the way to Pensacola. $475 for a previous plane trip to Pensacola to look for an apartment. $175 in temporary storage costs for his collection of sports memorabilia. Required: If the government reimburses him $900, how much, if any, may Reginald take as a moving expense deduction on his 2022 tax return?
Answers
Therefore , the solution of the given problem of unitary method comes out to be $500.
An unitary method is what?This common convenience, already-existing variables, or all important elements from the original Diocesan adaptable study that followed a particular methodology can all be used to achieve the goal. Both of the crucial elements of a term affirmation outcome will surely be missed if it doesn't happen, but if it does, there will be another chance to get in touch with the entity.
Here,
Reginald's total moving costs must first be determined in order to determine the moving expense deduction he may claim on his 2022 tax return.
The full cost of relocating is:
=> $450 (gasoline) + $250 (truck rental) + $100 (tow trailer) + $90 (food) + $35 (Starbucks) + $300 (motel lodging) + $175 (storage)
=> $1,400
Reginald's moving expenditures were covered by the government for $900, so the following are his actual moving costs:
=> $1,400 - $900 = $500
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Please help with hard polynomial problem
Let [tex]$f(x)=(x^2+6x+9)^{50}-4x+3$[/tex], and let [tex]$r_1,r_2,\ldots,r_{100}$[/tex] be the roots of [tex]$f(x)$[/tex].
Compute [tex]$(r_1+3)^{100}+(r_2+3)^{100}+\cdots+(r_{100}+3)^{100}$[/tex].
Answers
The value of [tex](r_1+3)^{100}+(r_2+3)^{100}+...+(r_{100}+3)^{100}[/tex] is -1500.
What is function?
A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences.
Here the given function is,
[tex]f(x)=(x^2+6x+9)^{50}-4x+3[/tex]
For any r , [tex](x^2+6x+9)^{50}-4x+3[/tex] is satisfy, Then take f(x)=0 then
=> [tex](x^2+6x+9)^{50}-4x+3=0[/tex]
Take x=r then
=> [tex](r^2+6r+9)^{50}-4r+3=0[/tex]
=> [tex](r^2+6r+9)^{50}=4r-3[/tex]
=> [tex]((r+3)^2)^{50}=4r-3[/tex]
=> [tex](r+3)^{100}=4r-3[/tex]
Then,
=> [tex]\sum_{i=1}^{100} (r_i+3)^{100}=\sum_{i=1}^{100} (4r_i-3)[/tex] = 4 × sum of roots - 300
Expanding [tex](x+3)^{100}-4x+3[/tex] using the binomial theorem, we get
=> f(x) = [tex]x^{100}+300x^{99}+....[/tex]
So sum of roots = -300 then
=> [tex](r_1+3)^{100}+(r_2+3)^{100}+...+(r_{100}+3)^{100} =[/tex] 4*(-300)-300=-1200-300
=> -1500.
Hence , The value of [tex](r_1+3)^{100}+(r_2+3)^{100}+...+(r_{100}+3)^{100}[/tex] is -1500.
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which of the following cases represent paired differences and which are differences between 2 independent samples?
Answers
In the given scenarios, Paired differences are option A, B, and D, the scenario which shows Differences between 2 independent samples is option C.
Paired differences refer to cases where the difference is measured within the same group of subjects or individuals, such as the difference in time spent with mother and father in a hetero family, or the difference in IQ between first-born and second-born twins.
Differences between 2 independent samples, on the other hand, refer to cases where the difference is measured between two separate groups, such as the difference in the number of books read per year by residents of California and New York.
Recognizing the type of difference is important in determining the appropriate statistical analysis to use in analyzing the data.
The answer option for paired are A, B, and D and for Differences between 2 independent samples is option C.
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_____The given question is incomplete, the complete question is given below:
Which of the following cases represent paired differences and which are differences between 2 independent samples?
A The difference between the average time a child spends with their mother and the average time the child spends with their father (in a hetero family) [Select]
B The difference between the average IQ of the first-born twin and of the second-born twin [ Select]
C The difference between the average number of books read per year by the residents of the states of California and New York [Select] The difference between the average amount grossed by Paramount Pictures and Warner Bros. movies [Select ]
D The difference between the shares of the republican and the democratic candidate in each year of the US elections. [ Select ]
Pls help !! I will mark brainilest
Answers
Answer:
m = -1
Step-by-step explanation:
may not be accurate, I haven't done this in a while
Answer:
-1y−y1=m(x−x1)
y−6=−1(x+5)
y−6=−1x+(−1×5)
y−6=−1x+−5
y−6=−1x−5
y=−1x−5+6
y=−1x+1
y=−x+1
m=−1
b=1
Step-by-step explanation: Hope this helps!! Mark me brainliest!
Hmm, try solving 13 to the power of 16 do it the long way :0
and just in case you do
2(2 + ab) + b(r + 3)
Answers
The answer of the given question based on solving 13 to the power of 16 is 13 to the power of 16 is 3,947,868,257,259,789. and the simplified expression is 7 + 2ab + br.
What is Expression?A expression is a combination of symbols or values that represents a particular concept or computation.
In mathematics, an expression is a combination of numbers, variables, operators, and/or functions that can be evaluated to produce a numerical result.
an expression is a way to represent an idea, computation, or meaning using a set of symbols or words.
To solve 13 to the power of 16, we can start by multiplying 13 by itself 16 times:
13 × 13 = 169
169 × 13 = 2197
2197 × 13 = 28,561
28,561 × 13 = 371,293
371,293 × 13 = 4,826,389
4,826,389 × 13 = 62,748,857
62,748,857 × 13 = 815,730,721
815,730,721 × 13 = 10,604,807,473
10,604,807,473 × 13 = 137,858,491,849
137,858,491,849 × 13 = 1,792,160,390,737
1,792,160,390,737 × 13 = 23,303,986,079,681
23,303,986,079,681 × 13 = 303,305,489,096,753
303,305,489,096,753 × 13 = 3,947,868,257,259,789
Therefore, 13 to the power of 16 is 3,947,868,257,259,789.
As for the second expression, we can simplify it using the distributive property of multiplication:
2(2 + ab) + b(r + 3) = 4 + 2ab + br + 3b
Simplifying further, we can combine the constant terms:
2(2 + ab) + b(r + 3) = 7 + 2ab + br
So the simplified expression is 7 + 2ab + br.
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factorise completely[tex]3x²-12xy
Answers
Answer:
3x(x - 4y)
Step-by-step explanation:
3x² - 12xy ← factor out 3x from each term
= 3x(x- 4y)
Mear's Taxi charges a $ 4.25 flat rate for a ride in the cab. In addition to that, they charge $ 0.47 per mile. Kevin has no more than $ 25 to spend on a ride. At most, how many miles can Kevin travel without exceeding his spending limit?
Answers
Answer: 44 miles
Step-by-step explanation:
Let's call the number of miles Kevin can travel "m". We can set up an equation using the given information:
4.25 + 0.47m ≤ 25
Solving for "m", we can begin by subtracting 4.25 from both sides:
0.47m ≤ 20.75
Then, divide both sides by 0.47:
m ≤ 44.15
Therefore, Kevin can travel at most 44 miles without exceeding his spending limit.
Three randomly selected children are surveyed. The ages of the children are 2, 4, and 12. Assume that samples of size n=2 are randomly selected
with replacement from the population of 2, 4, and 12. Listed below are the nine different samples. Complete parts (a) through (d).
2,2 2,4 2,12 4,2 4,4 4,12 12,2 12,4 12,12
a. Find the value of the population variance o²
Σ(x-1)
N
The formula for the population variance is o²=
where u is the population mean and N is the population size.
While either technology or the formula can be used to find the population variance, in this exercise, use technology. Determine the population
variance.
4
(Round to three decimal places as needed.)
Answers
Three youngsters are interviewed at random. Population variation is around [tex]18.67[/tex].
What is population standard deviation vs mean difference?The standard deviation refers to the square base of variance, which is the average of the squares departures from the mean. Both metrics capture distributional variability, although they use different measurement units: The units used to indicate standard deviation are the same as the values' original ones.
We compute population variance for what reason?In statistics, population standard deviation is a crucial indicator of dispersion. further reading. Statisticians compute variance to determine how order to overcome the drawbacks in a data gathering interact to one another. By calculating the population variance, one may also compute the dispersion in relation to the population means.
we need to first find the population mean [tex]u[/tex]
[tex]u = (2 + 4 + 12)/3 = 6[/tex]
To calculate the population variance
[tex]= [(-4)^{2} + (-2)^{2} + 6^{2} ]/3[/tex]
[tex]= (16 + 4 + 36)/3[/tex]
[tex]= 56/3[/tex]
[tex]= 18.67[/tex] (rounded to two decimal places)
Therefore, the population variance is approximately [tex]18.67[/tex].
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For which pair of functions is the exponential consistently growing at a faster rate than the quadratic over the interval 0 ≤ x ≤ 5?
Answers
One pair of functions that satisfies the given condition is:
Exponential function: [tex]f(x) = 1.46^x,[/tex] Quadratic function: [tex]g(x) = x^2[/tex]
What is expression ?In mathematics, an expression is a combination of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. Expressions can also include functions, brackets, and other symbols.
According to the given information:Let's consider the two functions:
Exponential function: [tex]f(x) = a^x, where a > 1[/tex]
Quadratic function: [tex]g(x) = x^2[/tex]
We want to find the pair of functions for which the exponential is consistently growing at a faster rate than the quadratic over the interval 0 ≤ x ≤ 5.
To determine this, we can compare the growth rates of the two functions by looking at their derivatives.
The derivative of the exponential function is:[tex]f'(x) = a^x * ln(a)[/tex]
The derivative of the quadratic function is: [tex]g'(x) = 2x[/tex]
To compare the growth rates of the two functions, we need to compare their derivatives. We want to find the value of x for which the exponential function is growing faster than the quadratic function, i.e., where f'(x) > [tex]g'(x).\\f'(x) > g'(x)\\a^x * ln(a) > 2x[/tex]
Now, we can solve for x:
[tex]a^x * ln(a) > 2xln(a)/2 * a^x > x[/tex]
Since we want to find the pair of functions for which the exponential is consistently growing at a faster rate than the quadratic over the interval 0 ≤ x ≤ 5, we need to find a value of a such that the inequality ln(a)/2 * [tex]a^5 > 5[/tex] is true for all values of a > 1.
We can use a graphing calculator or a numerical solver to find the value of a that satisfies this inequality. One possible solution is a ≈ 1.46.
Therefore, one pair of functions that satisfies the given condition is:
Exponential function: [tex]f(x) = 1.46^x,[/tex] Quadratic function: [tex]g(x) = x^2[/tex]
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What is the difference between the questionnaire and an interview?
Answers
Answer: Questionnaire refers to a research instrument, in which a series of question, is typed or printed along with the choice of answers, expected to be marked by the respondents, used for survey or statistical study. It consists of aformalisedd set of questions, in a definite order on a form, which are mailed to the respondents or manually delivered to them for answers. The respondents are supposed to read, comprehend and give their responses, in the space provided.
A ‘Pilot Study’ is advised to be conducted to test the questionnaire before using this method. A pilot survey is nothing but a preliminary study or say rehearsal to know the time, cost, efforts, reliability and so forth involved in it.
The interview is a data collection method wherein a direct, in-depth conversation between interviewer and respondent takes place. It is carried out with a purpose like a survey, research, and the like, where both the two parties participate in the one to one interaction. Under this method, oral-verbal stimuli are presented and replied by way of oral-verbal responses.
It is considered as one of the best methods for collecting data because it allows two way exchange of information, the interviewer gets to know about the respondent, and the respondent learns about the interviewer. There are two types of interview:
Personal Interview: A type of interview, wherein there is a face to face question-answer session between the interviewer and interviewee, is conducted.
Telephonic Interview: This method involves contacting the interviewee and asking questions to them on the telephone itself.
A water park sold 1679 tickets for a total of 44,620 on a warm summer day. Each adult ticket is $35 and each child ticket is $20. How many of each type of ticket was sold?
Answers
Therefore , the solution of the given problem of unitary method comes out to be the attraction sold 943 child tickets and 736 adult tickets on that particular day.
What is an unitary method?It is possible to accomplish the objective by using previously recognized variables, this common convenience, or all essential components from a prior malleable study that adhered to a specific methodology. If the expression assertion result occurs, it will be able to get in touch with the entity again; if it does not, both crucial systems will undoubtedly miss the statement.
Here,
Assume the attraction sold x tickets for adults and y tickets for kids.
Based on the supplied data, we can construct the following two equations:
=> x + y = 1679 (equation 1, representing the total number of tickets sold)
=> 35x + 20y = 44620 (equation 2, representing the total revenue generated)
Using the elimination technique, we can find the values of x and y.
When we divide equation 1 by 20, we obtain:
=> 20x + 20y = 33580 (equation 3)
Equation 3 is obtained by subtracting equation 2 to yield:
=> 15x = 11040
=> x = 736
When we enter x = 736 into equation 1, we obtain:
=> 736 + y = 1679
=> y = 943
As a result, the attraction sold 943 child tickets and 736 adult tickets on that particular day.
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Prove that for every real number if c is the root of a polynomial with rational coefficients then root ofa polynomial with integer coefficients: It may be helpful to suppose that is a solution to the polynomial equation: GnXn An-1xn-1_ +q1* + 4o Where qi € Q
Answers
Every real root of a polynomial with rational coefficients is also a root of a polynomial with integer coefficients.
Suppose that c is a root of the polynomial equation:
[tex]q_n[/tex] × [tex]x^n[/tex] + q_{n-1} × [tex]x^{n-1}[/tex] + ... + q1 × x + q0 = 0
where [tex]q_i[/tex] are rational coefficients. Since c is a root of this polynomial equation, we have:
[tex]q_n[/tex] × [tex]c^n[/tex] + [tex]q_{n-1}[/tex] × [tex]c^{n-1}[/tex] + ... + q1 × c + q0 = 0
Multiplying both sides of the equation by the common denominator of the coefficients [tex]q_i[/tex], we can obtain an equation with integer coefficients. Let d be the least common multiple of the denominators of the coefficients [tex]q_i[/tex]. Then we can write:
d × ([tex]q_n[/tex] × [tex]c^n[/tex] + [tex]q_{n-1}[/tex] × [tex]c^{n-1}[/tex] + ... + q1 × c + q0) = 0
Expanding the left-hand side of the equation, we obtain a polynomial with integer coefficients:
d × [tex]q_n[/tex] × [tex]x^n[/tex] + d × [tex]q_{n-1}[/tex] × [tex]x^{n-1}[/tex] + ... + d × q1 × x + d × q0 = 0
Since c is a root of the original polynomial equation, it is also a root of this polynomial with integer coefficients. Therefore, we have shown that if c is a root of a polynomial with rational coefficients, then it is also a root of a polynomial with integer coefficients.
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A cyclist rides off from rest, accelerating at a constant rate for 3 minutes until she reaches 40 kmh-1. She then maintains a constant speed for 4 minutes until reaching a hill. She slows down at a constant rate over one minute to 30 kmh-1. then continues at this rate for 10 minutes.
At the top of the hill she reduces her speed uniformly and is stationary 2 minutes later.
b
How far has the cyclist travelled? Its 9.75 km, but I don't understand how to get there
PLEASE SHOW YOUR WORK
Answers
Answer:
Step-by-step explanation:
To solve this problem, we need to use the equations of motion for constant acceleration, constant velocity, and constant deceleration. We'll break the problem into several parts and use these equations to find the distance traveled in each part. Then, we'll add up the distances to get the total distance traveled.
First, we need to convert the units of speed from km/h to m/s, since the equations of motion use meters per second. We have:
Initial speed (u) = 0 km/h = 0 m/s
Final speed (v) = 40 km/h = 11.11 m/s
Constant speed = 40 km/h = 11.11 m/s (for 4 minutes)
Final speed before hill = 30 km/h = 8.33 m/s
Speed at top of hill = 0 m/s
Acceleration (a) = (v-u)/t = (11.11-0)/(3*60) = 0.0611 m/s^2
PART 1: ACCELERATION PHASE
Time taken (t) = 3 minutes = 180 seconds
Distance traveled (s) = ut + (1/2)at^2
s = 0 + (1/2)0.0611(180^2) = 331.83 meters
PART 2: CONSTANT SPEED PHASE
Time taken (t) = 4 minutes = 240 seconds
Distance traveled (s) = vt
s = 11.11*240 = 2666.4 meters
PART 3: DECELERATION PHASE
Time taken (t) = 1 minute = 60 seconds
Deceleration (a) = (v-u)/t = (8.33-11.11)/60 = -0.0461 m/s^2 (negative since it's deceleration)
Distance traveled (s) = vt + (1/2)at^2
s = 8.3360 + (1/2)(-0.0461)*(60^2) = 494.7 meters
PART 4: CONSTANT SPEED PHASE
Time taken (t) = 10 minutes = 600 seconds
Distance traveled (s) = vt
s = 8.33*600 = 4998 meters
PART 5: DECELERATION PHASE TO STOP
Time taken (t) = 2 minutes = 120 seconds
Initial speed (u) = 8.33 m/s
Final speed (v) = 0 m/s
Deceleration (a) = (v-u)/t = (0-8.33)/120 = -0.0694 m/s^2
Distance traveled (s) = vt + (1/2)at^2
s = 8.33120 + (1/2)(-0.0694)*(120^2) = 733.3 meters
TOTAL DISTANCE TRAVELED:
Adding up the distances from each part, we get:
Total distance = 331.83 + 2666.4 + 494.7 + 4998 + 733.3 = 9184.23 meters = 9.18 km (rounded to two decimal places)
Therefore, the cyclist has traveled approximately 9.18 km.
For all values of x f(x) = 2x-3 and g(x) = x² + 2 (c) Solve fg(x) = gf(x)
Answers
Answer: x = 5 and x = 1.
Step-by-step explanation:
To solve fg(x) = gf(x), we need to find the expressions for fg(x) and gf(x) and then set them equal to each other.
fg(x) = f(g(x)) = f(x² + 2) = 2(x² + 2) - 3 = 2x² + 1
gf(x) = g(f(x)) = g(2x - 3) = (2x - 3)² + 2 = 4x² - 12x + 11
Now we set fg(x) equal to gf(x) and solve for x:
2x² + 1 = 4x² - 12x + 11
2x² - 12x + 10 = 0
Dividing both sides by 2 gives:
x² - 6x + 5 = 0
This quadratic equation factors as:
(x - 5)(x - 1) = 0
So the solutions are x = 5 and x = 1.
Therefore, the solutions to fg(x) = gf(x) are x = 5 and x = 1.
2 Sasha believes her soccer team plays better at away games than at home games. Her team played 12 games at home and 12 games away. She recorded the wins over this season. Based on these results, what is the probability for home and away wins? Is she right about her team playing better away? Show your work. Place: Home Away
Frequency: 8 7
Answers
Probability of winning at home is 0.67, Probability of winning away is 0.58.
How to calculate the probabilitiesTo calculate the probability of home and away wins, we need to divide the number of wins by the total number of games played at home and away respectively.
Probability of winning at home = (Number of wins at home) / (Total number of games played at home)
= 8 / 12
= 0.67
Probability of winning away = (Number of wins away) / (Total number of games played away)
= 7 / 12
= 0.58
Based on these results, Sasha's team has a higher probability of winning at home (0.67) compared to away games (0.58).
Therefore, her belief that her team plays better away is not supported by the data.
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(a) If a is a zero of the polynomial P(x), then must be a factor of P(x). (b) If a is a zero of multiplicity m of the polynomial P(x), then must be a factor of P(x) when we factor P completely.
(a) If a is a zero of the polynomial P(x), then must be a factor of P(x).
(b) If a is a zero of multiplicity m of the polynomial P(x), then must be a factor of P(x) when we factor P completely.
Answers
If a is a zero of the polynomial P(x), then (x-a) must be a factor of P(x) and [tex](x-a)^m[/tex] be a factor of P(x) when we factor P completely.
The values of x that fulfil the formula f(x) = 0 are the zeros of a polynomial. The polynomial's zeros are the x values for which the function's value, f(x), equals zero in this case. The degree of the equation f(x) = 0 determines how many zeros a polynomial has.
The locations when a polynomial equals 0 overall are known as its zeros. In layman's terms, we may state that a polynomial's zeros are variable values at which the polynomial equals 0. The zeros of a polynomial are often referred to as the equation's roots and are frequently written as,, and. A few techniques for locating polynomial zeros include grouping, factoring, and employing algebraic expressions.
(a) if we have zero at x=a of polynomial P(x)
then, (x-a) must be factor of P(x).
(b) if we have zero at x=a of polynomial P(x)
with multiplicity=m
then, [tex](x-a)^m[/tex] must be factor of P(x).
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Consider the line segment AB
shown. Which of the following
locations for point C makes ABC a right triangle with hypotenuse AB?
A - C(7,9)
B - C(1,4)
C - C(2,3)
D - C(8,7)
Answers
Consider the line segment AB shown the locations for point C makes ABC a right triangle with hypotenuse AB
C - C(2,3)How to find point C with hypotenuse ABIn a coordinate pair the points are as represented as (x, y).
The point that forms the right triangle is located by tracing the point on the x axis of of the point A and the point on the y axis of the point B. This is done below
A (2, 1) point on x axis here is 2B (9, 3) point on y axis here is 3therefore we can say that the point C that forms the right triangle is
C (2, 3)
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t/f the mean and standard deviation are more accurate measures of center and spread when the data is skewed
Answers
When the data is skewed, it is untrue that the mean and standard deviation are better indicators of the centre and spread.
The median is a better tool to use to locate the centre when it is skewed right or left with high or low outliers. The IQR is the most accurate indicator of spread when the median is the centre. When the mean is the centre, the standard deviation should be utilised because it gauges how far a data point is from the mean.
The standard deviation will be greatly overstated in cases when the distribution of the data is highly skewed, making it a poor choice as a measure of variability.
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You have a map that is missing a scale. The distance from Point A to Point B is
five inches on the map, and after driving it, you know it is 250 miles in reality.
Answers
The scale of the map in the question is 1 inch = 50 miles.
What is the scale of the map?A scale in a map is a relation that tells us how many units each unit in the map represents. In this case, we know that the distance between two points A and B on the map is 5 inches, while the actual distance between these two places is 250 miles.
Then we start with the relation:
5 inches = 250 miles.
But to get the scale of the map we need to see how many miles one inch represents in the map, then we can divide both sides of the equation by 5 to geT:
5 in = 250 mi
1 in = 250mi/5
1 in = 50 mi
The scale of the map is 1 inch to 50 miles.
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What’s the answer???
Answers
The difference in price between the two shops for 300 cm of ribbon is £3.60 - £2.40 = £1.20. Therefore, the answer is £1.20.
How to solve and what is Selling?
We can use proportions to find the cost of 300 cm of ribbon at each shop, and then subtract the cost at Shop B from the cost at Shop A to find the difference in price:
For Shop A:
140 cm of ribbon cost £1.68, so 1 cm of ribbon cost £1.68/140 = £0.012.
Therefore, 300 cm of ribbon would cost £0.012 x 300 = £3.60.
For Shop B:
215 cm of ribbon cost £1.72, so 1 cm of ribbon cost £1.72/215 = £0.008.
Therefore, 300 cm of ribbon would cost £0.008 x 300 = £2.40.
The difference in price between the two shops for 300 cm of ribbon is £3.60 - £2.40 = £1.20. Therefore, the answer is £1.20.
Selling is the process of exchanging goods or services for money or other valuable consideration. In business, selling is an essential part of the marketing and sales process, and involves identifying potential customers or clients, communicating with them about the features and benefits of the product or service being sold, and negotiating a price or other terms of the sale.
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