We have shown that KLMN has four congruent sides and opposite sides are parallel and perpendicular, which makes KLMN a square.
What is a graph?
A diagram (such as a series of one or more points, lines, line segments, curves, or areas) that represents the variation of a variable in comparison with that of one or more other variables.
To determine if KLMN is a square, we need to check if all four sides are congruent and if opposite angles are congruent.
We can find the length of each side using the distance formula:
KL = sqrt((10 - 6)^2 + (-5 - 0)^2) = sqrt(16 + 25) = sqrt(41)
LM = sqrt((5 - 10)^2 + (-9 - (-5))^2) = sqrt(25 + 16) = sqrt(41)
MN = sqrt((1 - 5)^2 + (-4 - (-9))^2) = sqrt(16 + 25) = sqrt(41)
NK = sqrt((6 - 1)^2 + (0 - (-4))^2) = sqrt(25 + 16) = sqrt(41)
So all four sides have the same length of sqrt(41), which is a necessary condition for a square.
To check if opposite angles are congruent, we can find the slope of each pair of opposite sides:
slope KL = (-5 - 0)/(10 - 6) = -5/4
slope MN = (-4 - (-9))/(1 - 5) = 5/4
slope LM = (-9 - (-5))/(5 - 10) = -4/5
slope NK = (0 - (-4))/(6 - 1) = 4/5
We can see that the slopes of KL and MN are negative reciprocals, as are the slopes of LM and NK. This means that opposite sides are parallel and perpendicular, respectively, which is another necessary condition for a square.
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We have demonstrated that KLMN is a square since it has four congruent sides and opposite sides that are parallel and perpendicular.
What is graph?
A visual representation of the variation of one variable with relation to one or more other variables, such as a collection of dots, lines, line segments, curves, or regions.
To determine if KLMN is a square, we need to check if all four sides are congruent and if opposite angles are congruent.
We can find the length of each side using the distance formula:
KL = sqrt((10 - 6)² + (-5 - 0)²) = sqrt(16 + 25) = sqrt(41)
LM = sqrt((5 - 10)² + (-9 - (-5))²) = sqrt(25 + 16) = sqrt(41)
MN = sqrt((1 - 5)² + (-4 - (-9))²) = sqrt(16 + 25) = sqrt(41)
NK = sqrt((6 - 1)² + (0 - (-4))²) = sqrt(25 + 16) = sqrt(41)
So all four sides have the same length of sqrt(41), which is a necessary condition for a square.
To check if opposite angles are congruent, we can find the slope of each pair of opposite sides:
slope KL = (-5 - 0)/(10 - 6) = -5/4
slope MN = (-4 - (-9))/(1 - 5) = 5/4
slope LM = (-9 - (-5))/(5 - 10) = -4/5
slope NK = (0 - (-4))/(6 - 1) = 4/5
We can observe that the slopes of KL, MN, LM, and NK are all negative reciprocals. Another prerequisite for a square is that the opposing sides must be parallel and perpendicular, respectively.
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Pentagon A'B'C'D'E' is the image of pentagon ABCDE under a
5
dilation with a scale factor of
2
a
B
What is the length of segment AE?
units
As a result, segment AE's length matches that of pentagons ABCDE and A'B'C'D'E'. Even as length of AE is not explicitly stated, we are unable to infer it based on the information in the problem.
What does a 4 scale factor mean?The scale factor refers to the proportion of one figure's side length to the other figure's corresponding side length. This time, XYUV=123=4. The scaling factor is hence 4.
Each side of the pentagon A'B'C'D'E' is twice as long as the equivalent side of the pentagon ABCDE because the dilatation scale factor is 2. Therefore:
A'B' = 2AB
B'C' = 2BC
C'D' = 2CD
D'E' = 2DE
E'A' = 2EA
Segment AE, which adds up to the sum of the first pentagon's two sides, needs to be as long as possible.
AB + BC + CD + DE=AE
We can swap the value of A'B', B'C', C'D', and D'E' for AB, BC, CD, and DE, respectively, since the dilation scale factor is 2:
AE = A'B'/2 + B'C'/2 + C'D'/2 + D'E'/2
AE = (2AB)/2 + (2BC)/2 + (2CD)/2 + (2DE)/2
AB + BC + CD + DE=AE
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Given vectors V1, V2,73, one can always write the zero vector as a linear combination as 0v1 + 02 + 0ï3 7. An important question is whether or not you can do it without using coefficients that are all o. For vi V2 V3 - -B can you write 7 as a linear combination where the coefficients are not all o? If so, give such a linear combination. Use complete sentences when giving your final answer to support your work.
Yes, it is possible to write 7 as a linear combination of the given vectors V1, V2, and V3 with coefficients that are not all zero.
One such linear combination can be given as follows:7 = -3V1 + 4V2 + V3To prove that this is indeed a linear combination of the given vectors, we can substitute the given vectors and coefficients in the above expression and verify that it gives the desired result.
So, we have:RHS = -3V1 + 4V2 + V3= -3(2i - 3j + k) + 4(-i + 2j + 3k) + (i - j - k)= -6i + 9j - 3k - i + 8j + 12k + i - j - k= 7j + 8kSince this result matches with the given vector 7, we can conclude that 7 can be written as a linear combination of the given vectors with coefficients that are not all zero.Hence, the required linear combination is:7 = -3V1 + 4V2 + V3.
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You report a confidence interval to your boss but she says that she wants a narrower range. SELECT ALL of the ways you can reduce the width of the confidence interval. o Increase the sample size o Decrease the sample size o Increase the confidence level o Decrease the confidence level I
o ncrease the mean o Decrease the mean
We can reduce the width of the confidence interval by increasing the sample size, reducing the confidence level and decreasing the mean (A, D, and F)
What is a confidence interval?A confidence interval is an estimate of the interval that has a specified probability of including an unknown population parameter.
The purpose of a confidence interval is to estimate the true value of the population parameter being measured, such as a mean, a standard deviation, or a proportion.
In general, a wider confidence interval indicates more uncertainty about the estimate, while a narrower confidence interval suggests more precision in the estimate.
We want narrower confidence intervals to demonstrate more precision, as this allows us to draw more reliable conclusions about population parameters.
We cannot reduce the width of the confidence interval by increasing the sample size, increasing the confidence level or increasing the mean.
Thus, the correct options are a, d, and f.
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Jeff goes to a fast food restaurant and orders some tacos and burritos. He sees on the nutrition menu that tacos are 150 calories and burritos are 380 calories. If he ordered
11 items and consumed a total of 3030 calories, how many tacos and how many burritos did Jeff order and eat?
Answer:
Step-by-step explanation:
Let's say Jeff ordered x tacos and y burritos.
From the problem, we know that:
- The calorie count of one taco is 150 calories
- The calorie count of one burrito is 380 calories
- Jeff ordered a total of 11 items
- Jeff consumed a total of 3030 calories
We can use the information given to form a system of equations:
x + y = 11 (Jeff ordered a total of 11 items)
150x + 380y = 3030 (Jeff consumed a total of 3030 calories)
To solve this system, we can use substitution.
Rearranging the first equation, we get:
x = 11 - y
Substituting this value of x into the second equation, we get:
150(11 - y) + 380y = 3030
Expanding and simplifying:
1650 - 150y + 380y = 3030
230y = 1380
y = 6
So Jeff ordered 6 burritos.
Substituting this value of y back into the first equation, we get:
x + 6 = 11
x = 5
So Jeff ordered 5 tacos.
Therefore, Jeff ordered 5 tacos and 6 burritos.
Jamel said that the 9 in 129,082 is two times greater than the 9 in 127,694 because it is two place values to the left. Is he correct?
Answer:
No, Jamel is not correct. While the 9 in 129,082 is indeed two places to the left of the 9 in 127,694, it is not two times greater.
In fact, the value of a digit in a number is determined by its place value. Each place value to the left represents a value that is ten times greater than the previous place value. So, the 9 in the ten thousands place of 129,082 represents a value of 9 x 10,000 = 90,000.
Similarly, the 9 in the ten thousands place of 127,694 represents a value of 9 x 10,000 = 90,000 as well. Therefore, both 9s have the same value and are not different by a factor of two.
In summary, Jamel's reasoning is flawed and not correct. The value of a digit in a number is based on its place value, and not its position relative to other digits.
Convert the following repeating decimal to a fraction in simplest form.
16
Answer:
16/99
Step-by-step explanation:
so you turn into an algebraic equation
[tex]x[/tex] = 0.1616161616
100[tex]x[/tex]=16.161616
100[tex]x[/tex]-1[tex]x[/tex]=99[tex]x[/tex]
99[tex]x[/tex]=16.1616-0.1616
= 16
[tex]x[/tex]= [tex]\frac{16}{99}[/tex]
the diagram shows a 4cm*4cm*4cm cube. find the length of the diagonal AB
Answer:
Step-by-step explanation:
First find BC:
[tex]BC^2=BD^2+DC^2[/tex] (Pythagoras Theorem)
[tex]\rightarrow BC^2=4^2+4^2[/tex]
[tex]\rightarrow BC^2=32[/tex]
[tex]\rightarrow BC=\sqrt{32} cm[/tex]
Now find AB:
[tex]AB^2=AC^2+BC^2[/tex] (Pythagoras Theorem)
[tex]\rightarrow AB^2=4^2+(\sqrt{32}) ^2[/tex]
[tex]\rightarrow AB^2=16+32[/tex]
[tex]\rightarrow AB^2=48[/tex]
[tex]\rightarrow AB=\sqrt{48}=6.9cm[/tex]
Diagonal AB = 6.9cm
please help i need to hurry
Answer:
( 9, 1/7 )
Step-by-step explanation:
I/9x- y=6/7 equation (1)
1/18x-2y=3/14 equation (2)
9x(1)-18x(2) multiply equation (1 ) by 9 and (2 ) by 18 and subtract
them
we get y=1/7
then put the value of y=1/7 in ( 1 ) or (2)
1/9x-1/7=6/7 ⇒x=9
therefore the right answer is ( 9, 1/7 )
& Management Sciences Assignment March 2 Jacob's Pet Shop is a small pet shop that stocks pet food and accessories and offers a local delivery service. The owner, Jacob, employs a shop manager-two shop assistants and a driver for deliveries. Jacob's Pet Shop PET SHOP Figure 1: Money, goods, services, and factors of production flow from Jacob's business into the econo 3.2.1. Discuss how money, goods and services and factors of production flow between Jacob's Pet Shop, consumers who buy pet food and accessorie from the business and the government. 3.2.2. Draw a diagram that shows all these flows. 3.3 List TWO features that a closed economy will prohibit. TOTAL MARKS [50] (2X-
Answer: 3.2.1. Money, goods, services, and factors of production flow between Jacob's Pet Shop, consumers, and the government as follows:
Money flows from consumers to Jacob's Pet Shop when they purchase pet food and accessories, and from Jacob's Pet Shop to the government in the form of taxes.
Goods and services flow from Jacob's Pet Shop to consumers when they purchase pet food and accessories, and from the government to Jacob's Pet Shop in the form of contracts or grants for the delivery service.
Factors of production flow from Jacob's Pet Shop to the economy in the form of employment opportunities for the shop manager, shop assistants, and driver, and from the economy to Jacob's Pet Shop in the form of raw materials and supplies for the business.
3.2.2. The diagram below illustrates the flow of money, goods and services, and factors of production between Jacob's Pet Shop, consumers, and the government:
+--------------------+
| |
| Consumers |
| |
+--------+-----------+
| Purchase pet food and accessories
+--------v-----------+
| |
| Jacob's Pet Shop |
| |
+-------+--------------+
| Sell pet food and accessories
|
+---------v-----------+
| |
| Government |
| |
+---------------------+
3.3. Two features that a closed economy will prohibit are:
International trade: A closed economy does not engage in trade with other countries. All economic activity is confined within the borders of the country.
Movement of capital: A closed economy does not allow the free movement of capital, such as investments or loans, in and out of the country. All capital transactions are restricted within the economy.
was this worth 5 points no way.
Step-by-step explanation:
Write the expression in complete factored form.
n7(n + 8) - 4(n + 8) = help
Answer:
Step-by-step explanation:
To factor the left side of the equation, we can first factor out the common factor of (n+8):
n7(n + 8) - 4(n + 8) = (n + 8)(n7 - 4)
So the complete factored form of the expression is (n + 8)(n7 - 4).
For accounting purposes, depreciation is: a. a decline in value of an asset. b. an allocation of a cost of an asset. c. the selling price of an asset
For accounting purposes, depreciation is an allocation of a cost of an asset.
What is depreciation?Both depreciation and amortization are techniques for spreading out an asset's cost throughout its useful life, although they are employed for various asset categories. Whereas amortisation is used for intangible assets like patents, copyrights, and trademarks, depreciation is utilised for tangible assets like buildings, machinery, and equipment. Depreciation is a method used to account for an asset's decline in value as a result of damage or obsolescence. During the course of the asset's useful life, the asset's cost is distributed to match the income it produces.
Depreciation is an accounting technique used to distribute a tangible asset's cost over the course of its useful life.
Hence, for accounting purposes, depreciation is an allocation of a cost of an asset.
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What is the average weight of an apple in grams, if 12 apples were bought for $6.72 and the apples were priced at $3.50/kg?
pls show working
Answer: The weight of the Apples is 160g
Step-by-step explanation:
If the apples were priced at $3.50/kg, then the cost of 1 kg of apples would be $3.50.
To find out the weight of 12 apples, we can divide the total cost of $6.72 by the cost of 1 kg of apples:
$6.72 ÷ $3.50/kg = 1.92 kg
So, the weight of 12 apples is 1.92 kg.
To find the average weight of 1 apple, we can divide the total weight of the 12 apples by the number of apples:
1.92 kg ÷ 12 = 0.16 kg
Finally, we need to convert the weight of 1 apple from kilograms to grams:
0.16 kg x 1000 g/kg = 160 g
Therefore, the average weight of an apple in grams is 160 g.
Jose and his children went into a grocery store where they sell apples for $2.25 each and mangos for $1.25 each. Jose has $20 to spend and must buy at least 9 apples and mangos altogether. Also, he must buy at least 3 apples and at most 9 mangos. If � x represents the number of apples purchased and � y represents the number of mangos purchased, write and solve a system of inequalities graphically and determine one possible solution.
Answer: $15.25
Step-by-step explanation:
Let x be the number of apples purchased and y be the number of mangos purchased. Then we have the following constraints:
2.25x + 1.25y ≤ 20 (Jose has $20 to spend)
x + y ≥ 9 (Jose must buy at least 9 apples and mangos altogether)
x ≥ 3 (Jose must buy at least 3 apples)
y ≤ 9 (Jose can buy at most 9 mangos)
To solve this system of inequalities graphically, we can plot the relevant lines and shade the feasible region.
First, we graph the line 2.25x + 1.25y = 20 by finding its intercepts:
When x = 0, we have 1.25y = 20, so y = 16.
When y = 0, we have 2.25x = 20, so x = 8.89 (rounded to two decimal places).
Plotting these intercepts and connecting them with a line, we get:
| *
16 | *
|
| /
| /
|/
0 *--------*
0 8.89
Next, we graph the line x + y = 9 by finding its intercepts:
When x = 0, we have y = 9.
When y = 0, we have x = 9.
Plotting these intercepts and connecting them with a line, we get:
|
9 | *
|
| /
| /
|/
0 *--------*
0 9
Finally, we shade the feasible region by considering the remaining constraints:
x ≥ 3: This means we shade to the right of the line x = 3.
y ≤ 9: This means we shade below the line y = 9.
Shading these regions and finding their intersection, we get:
| *
16 | * |
| |
| / |
| / |
|/ |
9 *--------*---
| | /
| |/
| *
0 *--------*
3 8.89
The feasible region is the shaded triangle bounded by the lines 2.25x + 1.25y = 20, x + y = 9, and x = 3.
To find one possible solution, we can pick any point within the feasible region. For example, the point (4, 5) satisfies all the constraints and represents buying 4 apples and 5 mangos, which costs 4(2.25) + 5(1.25) = $15.25.
2 numbers add together to make -4 but they subtract to make 8
Answer:
2=x y=-6
Step-by-step explanation:
x+y=-4
x-y=8
(-)(-)=+
(-)(+)=-
2+-6=-4
2--6=8
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One number is 13 less than another number. Let x represent the greater number. What is the sum of these two numbers?
Answer:
2x - 13
Step-by-step explanation:
If x represents the greater number, then the other number is x - 13. The sum of these two numbers is:
x + (x - 13) = 2x - 13
Suppose that the insurance companies did do a survey. They randomly surveyed 400 drivers and found that 320 claimed they always buckle up. We are interested in the population proportion of drivers who claim they always buckle up.a.i. x = __________ii. n = __________iii. p′ = __________b. Define the random variables X and P′, in words.c. Which distribution should you use for this problem? Explain your choice.d. Construct a 95% confidence interval for the population proportion who claim they always buckle up.i. State the confidence interval.ii. Sketch the graph.iii. Calculate the error bound.e. If this survey were done by telephone, list three difficulties the companies might have in obtaining random results.
We are interested in the population proportion of drivers who claim they always buckle upa.i. x = 320 ii. n = 400 iii. p′ = 0.8
b. The random variable X represents the number of drivers out of the sample of 400 who claim they always buckle up, while P′ represents the sample proportion of drivers who claim they always buckle up.
c. The distribution to use for this problem is the normal distribution because the sample size is large enough (n=400) and the population proportion is not known.
d. i. The 95% confidence interval for the population proportion who claim they always buckle up is (0.7709, 0.8291).
ii. The graph is a normal distribution curve with mean p′ = 0.8 and standard deviation σ = sqrt[p′(1-p′)/n].
iii. The error bound is 0.0291.
e. Three difficulties the insurance companies might have in obtaining random results from a telephone survey are:
Selection bias: The survey might not be truly random if the telephone numbers selected are not representative of the population of interest.
Nonresponse bias: People may choose not to participate in the survey or may not be reached, which could bias the results.
Social desirability bias: Respondents may give socially desirable answers rather than their true opinions, which could also bias the results.
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make t the subject of s=1/2*at^2
for h(x) = 4x-1, find h(0) and h(2)
Answer:
- 1 and 7
Step-by-step explanation:
to find h(0) substitute x = 0 into h(x)
h(0) = 4(0) - 1 = 0 - 1 = - 1
to find h2) substitute x = 2 into h(x)
h(2) = 4(2) - 1 = 8 - 1 = 7
Write the product in standard form.
(x - 7)²
Answer:
x² - 49
Step-by-step explanation:
(x - 7)² =
(x - 7) * (x - 7) =
x * x - 7 * 7 =
x² - 49
Whatis the value of # y_ When y = 5 and x = 10?
The value of y when x=36 is 31/3.
According to the question,
x varies directly with 3y+5
Thus,
[tex]x = k(3y+5)[/tex]
where K is any positive integer.
Now, from the question,
y=5/3 when x=10
Thus,
[tex]10 = k(3(\frac{5}{3} )+5)\\10 = k(10)\\k = 1[/tex]
Now we can use k to find the value of y when x=36:
[tex]36 = 1(3y+5)\\31 = 3y\\y = \frac{31}{3}[/tex]
In mathematics, an integer is a whole number that can be positive, negative, or zero. Integers are a fundamental concept in number theory and are used in a variety of applications across many fields. Integers can be represented using the symbol "Z" and are denoted by an integer value like -3, 0, 1, 2, 3, etc. They are used to count and measure quantities in real-life situations, such as the number of apples in a basket or the distance between two cities.
Integers are closed under addition, subtraction, and multiplication, meaning that if you add, subtract, or multiply any two integers, the result will also be an integer. However, the division is not always closed under integers, as dividing by zero is undefined, and dividing two integers can result in a non-integer value.
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Complete Question:
If x varies directly with 3y+5 and y=5/3 when x=10, then what is the value of y when x=36
three traffic lights on a street span 120 yards. if the second traffic light is 85 yards away from the first, how far in yards is the third traffic light from the seccond
The distance between the third traffic light from the second traffic light is 35 yards.
What is the distance?Three traffic lights on a street span 120 yards.
If the second traffic light is 85 yards away from the first.
Therefore, the third traffic light is 120 - 85 =35 yards away from the second traffic light.
This means that the third traffic light is directly adjacent to the first traffic light. The third traffic light from the second traffic light is at the same location as the first traffic light.
Thus, the distance between the third traffic light from the second traffic light is 35 yards.
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PLEASE HELP MARKING BRAINLEIST JUST ANSWER ASAP AND BE CORRECT
PLEASE HELP MARKING BRAINLEIST JUST ANSWER ASAP AND BE CORRECT
Answer:
To find the perimeter, we need to add up the lengths of all the sides. In this case, we have four sides with lengths given by the expressions 2v-5, 4v+5, 2v-5, and 8v-7.
So the perimeter P is:
P = (2v-5) + (4v+5) + (2v-5) + (8v-7)
Simplifying and combining like terms, we get:
P = 16v - 12
Therefore, the perimeter is 16v - 12.
Quadratic form. Suppose A is an n × n matrix and x is an n-vector. The triple product n' Az, a 1 x1 matrix which we consider to be a scalar (i.e., number), is called a quadratic form of the vector x, with coefficient matrix A. A quadratic form is the vector analog of a quadratic function au2, where a and u are both numbers. Quadratic forms arise in many fields and applications. (a) Show that a Az -Xij-1 Aijziz,j (b) Show that x"(AT)a - Ax. In other words, the quadratic form with the trans posed coefficient matrix has the same value for any a. Hint. Take the transpose of the triple product x Ax (c) Show that ((A AT)/2)-xA. In other words, the quadratic form with coefficient matrix equal to the symmetric part of a matrix (i.e., (A + AT)/2) has the same value as the original quadratic form. (d) Express 2c - 3rjt2- z2 as a quadratic form, with symmetric coefficient matrix A.
The quadratic form are a) ′ = ᵢⱼ∑ ᵢⱼᵢ, b) ′()=′, c) ((+)/2)=′, d) 2−3²−².
The question is asking for an expression of the quadratic form with coefficient matrix A for 2c-3rjt2-z2.
For (a): The quadratic form of the vector x with coefficient matrix A is expressed as ′ = ᵢⱼ∑ ᵢⱼᵢ, where is an × matrix and is an -vector.
For (b): Taking the transpose of the triple product ′ we get ′()=′, which shows that the quadratic form with the transposed coefficient matrix has the same value for any a.
For (c): The quadratic form with coefficient matrix equal to the symmetric part of a matrix (i.e., (A+AT)/2) is expressed as ((+)/2)=′, which shows that the quadratic form with coefficient matrix equal to the symmetric part of a matrix has the same value as the original quadratic form.
For (d): The expression 2c-3rjt2-z2 as a quadratic form, with symmetric coefficient matrix A, is expressed as ((+)/2)= 2−3²−², where is a symmetric matrix.
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Sharon is saving for a new phone. She starts with $42 and earns $15 each week from her parents. How many weeks will it take to save a total of $207?
Answer:
11 weeks
Step-by-step explanation:
42 + 15 * x = 207
42 + 15x = 207
15x = 207 - 42
15x = 165
x = 11
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The graph shows the velocity, v metres per second, of a car at time t seconds. Work out an estimate for the distance the car travelled for the first 8 seconds. Use 4 strips of equal width. -1-500- -1000- -500 0 V t
please help!!!
To estimate the distance traveled we need to find the area under the velocity-time graph from 0 to 8 seconds So,The estimate for the distance the car traveled for the first 8 seconds is 4000 meters.
Define velocity-time graph?A velocity-time graph is a graphical representation that shows the velocity of an object on the y-axis and time on the x-axis. It is used to depict the change in velocity over time and can provide information about the acceleration or deceleration of an object.
The height of each strip can be estimated by taking the average of the velocities at the beginning and end of the strip.
Using the trapezium rule, the estimated area of each strip is:
Strip 1: 0.5 x (0 + 2) x (0 + (-500)) = -500 m/s
Strip 2: 0.5 x (2 + 4) x (-500 + (-1000)) = -1500 m/s
Strip 3: 0.5 x (4 + 6) x (-1000 + (-500)) = -1500 m/s
Strip 4: 0.5 x (6 + 8) x (-500 + 0) = -500 m/s
The total estimated area is the sum of the areas of the 4 strips:
Total estimated area = -500 + (-1500) + (-1500) + (-500) = -4000 m/s
Since the area represents the distance traveled by the car, we can take the absolute value of the area to get the estimated distance traveled:
Estimated distance traveled is = |-4000| = 4000 meters
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Which of the following sets of numbers could not represent the three sides of a right triangle? O {35, 84, 91} O {5, 8, 10} O {33, 56, 65} O {42, 56, 70}
Answer: The numbers that cannot represent the three sides of a right triangle are {35, 84, 91}.
Step-by-step explanation:
The inverse sine, inverse cosine, and inverse tangent functions have the following domains and ranges. (Enter your answers in interval notation.)(a) The function sin−1 has domain and ranges(b) The function cos−1 has domain(c) The function tan−1 has domainand range
(a) The function sin⁻¹ has a domain of [-1, 1] and a range of [-π/2, π/2].
(b) The function cos⁻¹ has a domain of [-1, 1] and a range of [0, π].
(c) The function tan⁻¹ has a domain of (-∞, ∞) and a range of (-π/2, π/2).
In mathematics, the domain and range are concepts used to describe the inputs and outputs of a function.
The domain of a function is the set of all possible inputs or values of the independent variable for which the function is defined or meaningful.
The range of a function, on the other hand, is the set of all possible outputs or values of the dependent variable that the function can produce for all the inputs in the domain.
The inverse sine function, denoted sin⁻¹, takes as input a value between -1 and 1, and returns the angle whose sine is equal to that value. Therefore, its domain is [-1, 1], since the sine of an angle cannot exceed these limits. The range of sin⁻¹ is the set of all angles whose sine lies between -1 and 1. The smallest such angle is -π/2, which corresponds to a sine of -1, and the largest such angle is π/2, which corresponds to a sine of 1. Hence, the range of sin⁻¹ is [-π/2, π/2].
The inverse cosine function, denoted cos⁻¹, is similar to sin⁻¹, but takes as input a value between -1 and 1 and returns the angle whose cosine is equal to that value. Its domain is also [-1, 1], since the cosine of an angle cannot exceed these limits. However, the range of cos⁻¹ is different from that of sin⁻¹. It starts at 0, which corresponds to a cosine of 1, and ends at π, which corresponds to a cosine of -1. The reason for this difference is that the cosine of an angle is always positive or zero when the angle lies between 0 and π radians.
The inverse tangent function, denoted tan⁻¹, takes as input any real number and returns the angle whose tangent is equal to that number. Its domain is (-∞, ∞), since the tangent of an angle can take on any real value. The range of tan⁻¹ is between -π/2 and π/2, since the tangent of an angle becomes infinite at these values.
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Write each expression as a single power of 10.
A. 10-2. 10-4
B. 106 10-1
104
107
.
C.
D. (10-3)4
10-8
E.
106
D can be written as a single power of [tex]10: 10^{(-12).[/tex]
E. E is already a single power of [tex]10: 10^6.[/tex]
What are expressions, exactly?A term may be a number, a variable, the prοduct οf twο οr mοre variables, οr the prοduct οf a number and a variable. An algebraic expressiοn can cοnsist οf a single term οr a cοllectiοn οf terms. Fοr example, in the expressiοn 4x + y, the twο terms are 4x and y.
A. Since the base is the same, we can add the exponents of 10 to simplify 10(-2) * 10(-4). That is to say:
[tex]10^(-2) * 10^{(-4)} = 10^{(-2-4)} = 10^{(-6) (-6)[/tex]
As a result, A can be written as a single power of ten: 10 (-6).
B. To simplify (106 * 10(-1)) / 104 * 107, first simplify the numerator and denominator separately, then divide:
[tex](10^6 * 10^{(-1)}) / 10^4 * 10^7 = 10^{(6-1)} / 10^{(4-7)}= 10^5 / 10^{(-3)} = 10^{(5+3)} = 10^8[/tex]
As a result, B can be written as a single power of ten: 1008.
C. To simplify (104 * 107) / (103)4, we can start with the denominator:
[tex](10^4 * 10^7) / (10^3)^4 = (10^4 * 10^7) / 10^{12[/tex]
The exponents of 10 can then be added:
[tex](10^4 * 10^7) / 10^{12} = 10^{(4+7-12)} = 10^{(-1)[/tex]
As a result, C can be written as a single power of ten: 10 (-1).
D. To simplify (10(-3)),
We can multiply the exponents of 10 by 4:
[tex](10^{(-3)})^4 = 10^{(-3*4)} = 10^{(-12)[/tex]
As a result, D can be written as a single power of ten: 10 (-12).
E already has a single power of ten: 106.
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It takes 120 unit cubes to completely fill a rectangular prism. The prism is 10 unit cubes high. What are the other possible dimensions of the rectangular prism?
the possible dimensions of the rectangular prism are:
1 unit by 12 units by 10 units
2 units by 6 units by 10 units
3 units by 4 units by 10 units
Let the length and width of the rectangular prism be represented by l and w, respectively. Since the height of the prism is 10 unit cubes, we know that the volume of the prism is:
V = l × w × 10
We are also given that the prism requires 120 unit cubes to fill completely, which means:
V = l × w × 10 = 120
Dividing both sides by 10, we get:
l × w = 12
Now we need to find pairs of positive integers l and w that multiply to 12. These are:
1 × 12
2 × 6
3 × 4
Therefore, the possible dimensions of the rectangular prism are:
1 unit by 12 units by 10 units
2 units by 6 units by 10 units
3 units by 4 units by 10 units
Note that these dimensions are not unique, as we can also obtain different dimensions by exchanging the length and width, or by rotating the prism.
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Enter the values needed to find the
length BC. (Simplify your answer.)
A (-5x, 4y)
B (-2x, -4y)
BC=√([?])² + (3y)²
C (7x, -1y)
Distance Formula
d = √√(x₂ − ×₁)² + (y₂ − y₁)²
The missing value to find the length of BC is 9x.
What is distance formula?The distance formula is a formula for calculating the separation in coordinates between two places. It is provided by and deduced from the Pythagorean theorem by:
distance = √((x₂ - x₁)² + (y₂ - y₁)²)
The distance formula is used to compute distances between objects or places in many disciplines, including geometry, physics, and engineering.
The distance formula is given as:
distance = √((x₂ - x₁)² + (y₂ - y₁)²)
Substituting the values of the coordinates of B and C we have:
distance = √((7x - (-2x))² + (-1y - (-4y))²)
distance = √((9x)² + (3y)²)
distance = √(81x² + 9y²)
distance = 3√(9x² + y²)
Hence, the missing value to find the length of BC is 9x.
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