Step-by-step explanation:
1)561
2)5
3)40
4)90
5)3
6)500
7)200
8)100
9)5
10)50
What is the value of 3x + 6 if x = -5
Answer:
-9
Step-by-step explanation:
x = -5
3x + 6
Since x = -5..
Do this
3(-5) + 6
Perform
-15 + 6
Answer: -9
Therefore, when x is equal to -5, the value of 3x + 6 is -9.
What is equation?An equation is a statement that expresses the equality of two mathematical expressions using mathematical symbols such as variables, numbers, and mathematical operations. The equality is represented by an equal sign "=" between the two expressions. Equations are used to represent mathematical relationships and solve problems in various fields such as physics, chemistry, engineering, and economics.
Given by the question.
3x + 6 = 3(-5) + 6
= -15 + 6
= -9
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Kate is x years old. Lethna is 3 times as old as Kate. Mike is 4 years older than Lethna. write down an expression, in terms of x for Mike's age
Answer: Mike is ( 3x + 4 ) years old
Step-by-step explanation:
K -> x y/o
L -> 3x y/o
M -> (3x + 4) y/o
A square is inscribed in a right triangle with leg lengths 6 and 8 so that they have a common right angle. FInd the square's side length.
Answer:
10 units
Step-by-step explanation:
Here, legs = base and perpendiculars.
So, Clearly given Base = 6 units Perpendicular = 8 cm
Square's Side = Hypotenuse.
By Pythagoras theorem,
H² = B²+P²
H ² = 6²+8²
H² = 36+64 = (10)²
H = 10 units.
Square's Side length = 10 units
A fruit basket holds x2 x2 – 3 x x + 12 apples. Maha takes out 4 x− x− 6 of them. How many apples are left in the basket?
p.s Please explain how you solve this question!
The answer of the given question based on the given equation that fruit basket holds x2 x2 – 3 x x + 12 apples and Maha takes out 4 x− x− 6 of them the answer is , there are x² - 6x + 18 apples left in the basket after Maha takes out 4x - x - 6 of them.
What is Expression?In mathematics, expression is combination of numbers, variables, and mathematical operations that can be evaluated to produce value. An expression can be as simple as single number or variable, or it can be complex, involving many different operations and variables.
Expressions can be used to represent many different mathematical ideas, like equations, inequalities, functions, and more. They can be used to model real-world situations, make predictions, and solve problems in wide variety of fields, like physics, economics, engineering, and more.
The fruit basket initially holds x² - 3x + 12 apples. If Maha takes out 4x - x - 6 of them, then we can subtract this expression from the initial number of apples to find how many are left in the basket.
So, the expression for the number of apples left in the basket is:
x² - 3x + 12 - (4x - x - 6)
We can simplify this by combining like terms:
x² - 3x + 12 - 4x + x + 6
Simplifying further:
x² - 6x + 18
Therefore, there are x² - 6x + 18 apples left in the basket after Maha takes out 4x - x - 6 of them.
It's important to note that we cannot simplify 4x - x - 6 to 3x - 6 in this case because the two terms have different coefficients. Instead, we need to distribute the negative sign to both terms inside the parentheses to get 4x - x - 6 = 4x - 1x + (-6) = 3x - 6, which we can then use in the expression for the number of apples left in the basket.
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This prism has a right triangle for a base. The volume of the prism is
54 cubic units. What is the value of h?
A
a
CA
6.
OF
'В
mi
E
Answer:
9
Step-by-step explanation:
The base of the prism is a right triangle.
The hypotenuse measures 5. One leg measures 4.
The other leg must measure 3 since it is the Pythagorean triple 3, 4, 5.
Area of the base = 3 × 4 / 2 = 6
V = Bh
54 = 6 × h
h = 9
HELP ASAP PLEASE!! A yard plan includes a rectangular garden that is surrounded by bricks. In the drawing, the garden is 7 inches by 4 inches. The length and width of the actual garden will be 35 times larger than the length and width in the drawing.
What is the perimeter of the drawing? Show your work.
What is the perimeter of the actual garden? Show your work.
What is the effect on the perimeter of the garden with the dimensions are multiplied by 35? Show your work.
Answer:
(a) perimeter of the drawing = (7 x2) + (4 x 2)
= 22 inches
(b) perimeter of the actual garden = 2(7 x 35) + 2(4 x 35)
= 490 + 280
= 770 inches
(c) 22 x 35 = 770
i don't really understand the last question. sorry if i gave you the wrong answers
a jar contains 6 red marbles numbered 1 to 6 and 12 blue marbles numbered 1 to 12. a marble is drawn at random from the jar. find the probability of the given event. write your answers as reduced fractions. (a) the marble is red your answer is : (b) the marble is odd-numbered
a) The total number of marbles is 6 + 12 = 18.P(a red marble) = 6/18 = 1/3
b) There are 12 odd-numbered marbles in total. P(an odd-numbered marble) = 12/18 = 2/3
The probability that the marble drawn from the jar is red can be found using the formula for probability. The probability formula can be written as the ratio of the number of favorable outcomes to the total number of possible outcomes. P(a red marble) = number of red marbles / total number of marbles. In this case, there are 6 red marbles and 12 blue marbles in the jar. Therefore, the total number of marbles is 6 + 12 = 18.P(a red marble) = 6/18 = 1/3(b) The probability that the marble drawn from the jar is odd-numbered can be found using the formula for probability. The probability formula can be written as the ratio of the number of favorable outcomes to the total number of possible outcomes. P(an odd-numbered marble) = number of odd-numbered marbles / total number of marbles. In this case, there are 6 red marbles numbered 1 to 6 and 12 blue marbles numbered 1 to 12 in the jar. Therefore, the total number of marbles is 6 + 12 = 18.To find the number of odd-numbered marbles, we need to count the number of red and blue marbles numbered 1, 3, 5, 7, 9, 11. There are 6 odd-numbered red marbles and 6 odd-numbered blue marbles. Therefore, there are 12 odd-numbered marbles in total. P(an odd-numbered marble) = 12/18 = 2/3
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formation about a sample is given. Assume that the sampling distribution is symmetric and bell-shaped. P_1 - P_2 = 0.15 and the margin of error for 95% confidence is 5%. (a) Indicate the parameter being estimated.(b) Use the information to give a 95% confidence interval.
(a) Parameter being estimated in the given information is the difference between two proportions (p_1 - p_2).
(b) A 95% confidence interval is given by (0.075, 0.225)
(a) The parameter being estimated is the difference between two population proportions, which is denoted by (p_1 - p_2).
(b) The margin of error for a 95% confidence interval is 5%, which means that the critical value of z is 1.96 (obtained from a standard normal distribution table). Using the formula for the margin of error, we can write:
1.96 * √(p_1_hat*(1-p_1_hat)/n_1 + p_2_hat*(1-p_2_hat)/n_2) = 0.05
where p_1_hat and p_2_hat are the sample proportions from the two samples, and n1 and n2 are the sample sizes.
Solving for p_1_hat - p_2_hat, we get:
p1_hat - p2_hat = ±0.075
Since we are interested in a 95% confidence interval, we can subtract and add this value from P1 - P2 to obtain the interval:
P_1 - P_2 ± 0.075
Substituting the given value of P_1 - P_2 = 0.15, we get:
95% Confidence Interval: (0.075, 0.225)
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Thabang save money by putting coins in a money box. The money box has 600 coins that consist of 20 cents and 50 cents
So calculate how many 50 cents pieces are in the container if there are 220 pieces of 20 cents
The number of 50 cents in the container is 380 fifty cents
How to find the number of 50 cents in the container?Since Thabang save money by putting coins in a money box. The money box has 600 coins that consist of 20 cents and 50 cents
To calculate how many 50 cents pieces are in the container if there are 220 pieces of 20 cents, we proceed as follows.
Let
x = number of 20 cents and y = number of 50 centsSince the total number of cents in the container is 600, we have that
x + y = 600
So, making y subject of the formula, we have that
y = 600 - x
Since x = 220
y = 600 - 220
= 380
So, there are 380 fifty cents
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Use the traditional square of opposition to determine whether the following immediate inferences are valid or invalid. Name any fallacies that are committed.
All advocates of school prayer are individuals who insist on imposing their views on others.
Therefore, some advocates of school prayer are individuals who insist on imposing their views on others
No fallacy is committed in the given immediate inference, as it follows the rules of the square of opposition.
The immediate inference presented in the statement is valid and can be categorized as a particular affirmative proposition (I-type) of the square of opposition.According to the square of opposition, a particular affirmative proposition can be inferred from a universal affirmative proposition (A-type) when the subject is distributed.
In the given statement, the universal affirmative proposition is "All advocates of school prayer are individuals who insist on imposing their views on others," which distributes the subject "advocates of school prayer."
Therefore, the particular affirmative proposition "Some advocates of school prayer are individuals who insist on imposing their views on others" can be inferred from the universal affirmative proposition.
No fallacy is committed in the given immediate inference, as it follows the rules of the square of opposition. However, it should be noted that the validity of the inference does not necessarily imply the truthfulness of the statement, as it is possible for some advocates of school prayer to not insist on imposing their views on others.
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A student takes a multiple-choice test that has 10 questions. Each question has four choices. The student guesses randomly at each answer. Round the answers to three decimal places Part 1 of2 (a) Find P(5) P(5)- Part 2 of2 (b) Find P(More than 3) P(More than 3)
A student attempts a 10-question multiple-choice test where each question presents four options, and the student makes random guesses for each answer. So the probability of (a) P(5)= 0.058 and (b) P(More than 3)= 0.093.
Part 1: Calculation of probability of getting 5 questions correct
(a) P(5)The formula used to find the probability of getting a certain number of questions correct is:
P(k) = (nCk)pk(q(n−k))
Where, n = total number of questions
(10)k = number of questions that are answered correctly
p = probability of getting any question right = 1/4
q = probability of getting any question wrong = 3/4
P(5) = P(k = 5) = (10C5)(1/4)5(3/4)5= 252 × 0.0009765625 × 0.2373046875≈ 0.058
Part 2: Calculation of probability of getting more than 3 questions correct
(b) P(More than 3) = P(k > 3) = P(k = 4) + P(k = 5) + P(k = 6) + P(k = 7) + P(k = 8) + P(k = 9) + P(k = 10)
P(k = 4) = [tex]10\choose4[/tex](1/4)4(3/4)6 = 210 × 0.00390625 × 0.31640625 ≈ 0.02
P(k = 5) = [tex]10\choose5[/tex](1/4)5(3/4)5 = 252 × 0.0009765625 × 0.2373046875 ≈ 0.058
P(k = 6) = [tex]10\choose6[/tex](1/4)6(3/4)4 = 210 × 0.0002441406 × 0.31640625 ≈ 0.012
P(k = 7) = [tex]10\choose7[/tex](1/4)7(3/4)3 = 120 × 0.00006103516 × 0.421875 ≈ 0.002
P(k = 8) = [tex]10\choose8[/tex](1/4)8(3/4)2 = 45 × 0.00001525878 × 0.5625 ≈ 0.001
P(k = 9) = [tex]10\choose9[/tex](1/4)9(3/4)1 = 10 × 0.000003814697 × 0.75 ≈ 0.000
P(k = 10) = [tex]10\choose10[/tex](1/4)10(3/4)0 = 1 × 0.0000009536743 × 1 ≈ 0
P(More than 3) = 0.020 + 0.058 + 0.012 + 0.002 + 0.001 + 0.000 + 0≈ 0.093
Therefore, the probabilities of the given situations are: P(5) ≈ 0.058, P(More than 3) ≈ 0.093.
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a credit risk study found that an individual with good credit score has an average debt of $15,000. if the debt of an individual with good credit score is normally distributed with standard deviation $3,000, determine the shortest interval that contains 95% of the debt values.
The shortest interval that contains 95% of the debt values is $9,492.02 to $20,507.98
How do we calculate the interval values?Given that a credit risk study found that an individual with good credit score has an average debt of $15,000 and the debt of an individual with good credit score is normally distributed with standard deviation $3,000.
Then the 95% confidence interval can be calculated as follows:
Upper limit: µ + Zσ
Lower limit: µ - Zσ
Where
µ is the mean ($15,000)Z is the z-scoreσ is the standard deviation ($3,000).The z-score corresponding to a 95% confidence interval can be found using the standard normal distribution table.
The area to the left of the z-score is 0.4750 and the area to the right is also 0.4750.
The z-score corresponding to 0.4750 can be found using the standard normal distribution table as follows:z = 1.96Therefore
Upper limit: µ + Zσ= $15,000 + 1.96($3,000) = $20,880
Lower limit: µ - Zσ= $15,000 - 1.96($3,000) = $9,120.02
The shortest interval that contains 95% of the debt values is $9,492.02 to $20,507.98.
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Find the missing side. Round your
answer to the nearest tenth.
15 m
32°
X
Answer:
x=9.4
using soh cah toa:
x=opposite side
15=adjacent side
using tan(toa)
[tex]tan32=\frac{x}{15}[/tex]
[tex]x=15tan32[/tex]
[tex]x=9.4[/tex]
An equilateral triangle as a side of 35 cm what is the distance around the triangle
Answer:
Step-by-step explanation:
To find the perimeter of an equilateral triangle is equalled to 3s.
35 x 3
105
7) Roy buys pizza for his friends. A whole pizza costs P 190. 00 and P 40. 00 for every
additional topping. If he spent P 1070 for pizza with 3 sets of additional toppings, how
many whole pizzas did he buy?
Roy purchased whole pizzas for P 190.00 each. To determine the number of whole pizzas he bought, we can divide the total cost of the pizzas by the cost of each pizza. Therefore, the calculation P 950.00 / P 190.00 results in 5, indicating that Roy bought five whole pizzas.
Roy spent a total of P 1070 for pizza with 3 sets of additional toppings. Since each set of additional toppings costs P 40.00, then the total cost of the toppings is 3 x P 40.00 = P 120.00. Subtracting this from the total amount spent gives us P 950.00, which is the cost of the pizzas alone.
Since each whole pizza costs P 190.00, we can divide the cost of the pizzas by the cost of each pizza to find the number of whole pizzas Roy bought. Therefore, P 950.00 / P 190.00 = 5.
Thus, Roy bought 5 whole pizzas.
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PLSS HELP IVE TRIED EVERYTHING
Answer:
Step-by-step explanation: To obtain the function g(x) that represents the indicated transformations of the graph of f(x) = 2, which are a translation 1 unit up followed by a horizontal stretch by a factor of 2, we can follow these steps:
To translate f(x) = 2 one unit up, we can add 1 to the function: f(x) + 1.
To horizontally stretch f(x) + 1 by a factor of 2, we can multiply the input (x) by 1/2: f(1/2 x) + 1.
Therefore, the function g(x) that represents the indicated transformations of f(x) is:
g(x) = f(1/2 x) + 1
g(x) = 2(1/2 x) + 1
g(x) = x + 1
Gail averages 153 points per bowling game with a standard deviation of 14.5 points. Suppose Gail's points per bowling game are normally distributed. Let X= the number of points per bowling game. Then X∼N(153,14.5). If necessary, round to three decimal places.
Suppose Gail scores 108 points in the game on Thursday. The z-score when x = 108 is __
. The mean is __
Gail scores 153 points on average every bowling game, with a 14.5 point standard deviation. Assume Gail's bowling game points are evenly divided. With x = 108, the mean is 153, and the z-score is -3.103.
The z-score when Gail scores 108 points in a game is calculated as:
z = (x - μ) / σ
where x = 108 is the observed score, μ = 153 is the mean, and σ = 14.5 is the standard deviation.
Plugging in the values, we get:
z = (108 - 153) / 14.5 ≈ -3.103
Rounding to three decimal places, the z-score when Gail scores 108 points in a game is approximately -3.103.
The mean is μ = 153, which is given in the problem statement.
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It is known that diskettes produced by a certain company will be defective with probability 0.01, independently of each other. The company sells the diskettes in packages of size 10 and offers a money-back guarantee that at most 1 of the 10 diskettes in the package will be defective.If someone buys 3 packages, what is the probability that he or she will return exactly 1 of 3 packages?
The probability of someone returning exactly 1 of the 3 packages can be calculated as:P(1 out of 3 packages is returned) = C(3, 1) × P(0 or 1 diskette is defective)¹ × (1 - P(0 or 1 diskette is defective))²P(1 out of 3 packages is returned) = C(3, 1) × (0.9043820371)¹ × (0.0956179629)²P(1 out of 3 packages is returned) = 0.2448700124Therefore, the required probability of someone returning exactly 1 of the 3 packages is 0.2448700124.
The given data from the question is that the company produces diskettes which have the probability of being defective as 0.01. The packages that are sold have a size of 10 and the guarantee says that there can be at most one defective diskette in the package. Now, the question is to find the probability of someone returning exactly 1 of the 3 packages that they have bought. So, the given data can be summarized as:Given:Probability of the diskette being defective, p = 0.01Guarantee: At most one diskette in the package of size 10 is defective.Now, let's solve the problem using probability theory
Probability of 1 diskette being defective in a package of size 10 can be calculated as:P(defective) = p = 0.01P(non-defective) = 1 - p = 0.99Using the given guarantee, probability of at most one defective diskette in a package of size 10:P(0 or 1 diskette is defective) = P(0 defective) + P(1 defective)P(0 or 1 diskette is defective) = C(10, 0) × (0.99)¹⁰ + C(10, 1) × (0.99)⁹ × (0.01)P(0 or 1 diskette is defective) = 0.9043820371Using the above probability
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In the diagram, PQRS ~ TUVW. Find the value of x .
The value of x is 8
What is a trapezoid?A quadrilateral with one pair of parallel sides. It is a typical mathematical shape that is employed in numerous disciplines. The area of a trapezoid is computed by multiplying the height by half of the sum of the lengths of the two bases.
Given that trapezoid PQRS ~ TUVW are similar;
So, VW/RS = UT/QP
x/12 = 6/9
x = 8
Therefore, the value of x is 8
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Trapezoid A trapezoid is a four-sided shape with one pair of parallel sides According to the question The value of x is 8
What is a trapezoid?A trapezoid is a four-sided shape with one pair of parallel sides. It has two non-parallel sides, or legs, and two parallel sides, or bases. The bases are usually of different lengths, and the sides are typically not equal. The angles of a trapezoid are not necessarily all the same size, but the two opposite angles are always equal. Trapezoids can be found in everyday objects like the shape of a deck, a rectangular door frame, or a sloped roof. Trapezoids can also be found in geometry, where they are used to calculate the area and perimeter of the shape.
Given that trapezoid PQRS ~ TUVW are similar;
So, VW/RS = UT/QP
x/12 = 6/9
x = 8
Therefore, the value of x is 8
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If an object is dropped from the top of a building, its position in feet above the ground is given by s(t)=−8t^2+288, where t is the time in seconds since it was dropped. What is the velocity when it hits the ground? a) Increasing 96 feet per second b)decreasing 16 feet per second c) 0 feet per second d)decreasing 96 feet per second
The velocity when the object hits the ground is 4.025 ft/s. The answer is: C) 0 feet per second.
If an object is dropped from the top of a building, its position in feet above the ground is given by s(t)=−8t^2+288, where t is the time in seconds since it was dropped. Given that an object is dropped from the top of a building,
then its initial velocity, u = 0 (since it was dropped and not thrown). We want to find the velocity when the object hits the ground, i.e., the time it takes the object to reach the ground. We know that the position of the object when it reaches the ground is s(t) = 0.
Therefore:−8t^2+288 = 0Solving for t, we get:−8t^2 = −288t^2 = 36t = sqrt(36) = ±6 sSince time cannot be negative, t = 6 s. Then the final velocity at impact, v, can be calculated as follows:v = u + at + s/v = 0 + (-32.2)(6) + (-8(6)² + 288)/v = -193.2 + (-288 + 288)/v = -193.2 + 0/v = -193.2/v = -193.2/-48v = 4.025 (rounded to three decimal places)
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The terminal ray of angle A, drawn in standard position, passes through the point (-4,
-6). What is the value of sec(A)?
Give your answer in simpliest radical form.
The value of sec A as required to be determined in the task content is; -√13 / 2.
What value represents sec A in the given scenario?As evident from the task content; it follows that the terminal ray of angle A, drawn in standard position, passes through the point (-4, -6).
Therefore, the length that the line from the origin to A has length;
L = √((-4)² + (-6)²)
L = √52.
On this note, it follows that the value of sec A which is represented by; hypothenuse/ adjacent is;
sec (A) = -√52 / 4
sec (A) = -√13 / 2.
Ultimately, the value of sec (A) as required is; -√13 / 2.
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Approximately 17.7% of vehicles traveling on a certain stretch of expressway exceed 110 kilometers per hour. If a state trooper randomly selects 154 vehicles and captures their speeds with a radar gun, what is the probability that at least 35 of the selected vehicles exceed 110 kilometers per hour?Use Excel to find the probability, rounding your answer to four decimal places.
Using Excel, the probability that at least 35 of the 154 randomly selected vehicles exceed 110 kilometers per hour when 17.7% of vehicles exceed this speed on the expressway is approximately 0.0027, rounded to four decimal places.
To solve this problem in Excel, we can use the binomial distribution function. In this case, the probability of success (a vehicle exceeding 110 kilometers per hour) is p = 0.177, and the number of trials (vehicles selected) is n = 154.
We want to find the probability of at least 35 successes (vehicles exceeding 110 kilometers per hour), which can be calculated using the formula:
=1-BINOM.DIST(34,154,0.177,TRUE)
This formula gives a probability of 0.0027, which is the probability that at least 35 of the selected vehicles exceed 110 kilometers per hour. Therefore, the answer is 0.0027, rounded to four decimal places.
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if you weigh 160 pounds, how many drinks in four hours would you need to drink to be definitely illegal?
According to the provided scenario, if you weigh 160 pounds, then 3 drinks in four hours would make you definitely illegal.
If a person weighs 160 pounds and drinks alcohol at a moderate rate, then after 3 drinks in four hours, their BAC (blood alcohol concentration) would be around 0.08, which is considered legally impaired and definitely illegal. However, it is important to note that this estimate is based on various factors such as the person's gender, age, and metabolism, and can vary from person to person.
Therefore, it is always advisable to drink responsibly and not drive after consuming alcohol.
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What is the area of the triangle?
Answer:
Step-by-step explanation:
Given:
Side a and Side b are 6 and 5.
The angle C is 131.
This is an obtuse scalene triangle as identified.
Area = ab * sin (C)/2 = 11.32064
choice matrix is shown. Complete the choice matrix by selecting the value equivalent to each function output.
Consider the functions shown.
f(x) = -3(2^x)
g(x) = -3 + 2x
h(x) = 2(3^x)
j(x) = -3 – 2x
Answer:
f(x) = -3(2^x)
g(x) = -3 + 2x
h(x) = 2(3^x)
j(x) = -3 – 2x
So,
f(2) = -3(2^2)
f(2) = -3(4)
f(2) = -12
g(-2) = -3 + 2(-2)
g(-2) = -3 -4
g(-2) = -7
h(2) = 2(3^2)
h(2) = 2(9)
h(2) = 18
j(-2) = -3 – 2(-2)
j(-2) = -3 –4
j(-2) = -7
PLEASE HELP!!
20 POINTS
A rental car company offers two rental plans, Plan A and Plan B, for the same economy size car. For both plans, the total rental cost f(m)
is a function of the number of miles m
that the car is driven.
Plan A: f(m)= 0.12+75
Plan B: f(m)+0.35
I. In complete sentences, translate each function into a verbal model describing the total cost of the rental in terms of the number of miles that the car is driven.
II. For each function, determine how the rate of change will affect the total cost of a car rental.
III. For a car rental that will include a maximum of 250 miles for the duration of the rental, which plan is the most cost effective?
Answer: At 250 miles, plan B is the most cost effective.
Step-by-step explanation:
Answers are under the questions below.
I. In complete sentences, translate each function into a verbal model describing the total cost of the rental in terms of the number of miles that the car is driven.
Plan A will charge .12 per mile plus a one time fee of $75.
Equation is y = .12x + 75 with the $75 being the y intercept or one time fee.
Plan B will charge .35 per mile with no one time fee.
Equation is y = .35x with the y intercept being (0) zero.
II. For each function, determine how the rate of change will affect the total cost of a car rental.
Initially the Plan A will be more expensive because of the one time fee, even though the rate is .12 per mile, which is much less than Plan B.
Plan B charges .35 per mile, but will eventually catch up in cost.
At approximately 326 miles the cost of each plan will equal at $114.13.
After 326 miles, Plan B will cost more than Plan A.
III. For a car rental that will include a maximum of 250 miles for the duration of the rental, which plan is the most cost effective?
Plan A, substitute 250 miles for x
Cost = .12 (250) + 75
Cost = $105
Plan B, substitute 250 miles for x
Cost = .35 (250)
Cost = $87.50
At 250 miles, plan B is the most cost effective, saving $17.50.
graph is attached.
Where is point c if c 2 units closer to b than it is a?
As per the points A, B, and C are colinear points and the point C lies between points A and B on the line.
Let's begin by finding the distance between points A and B. Using the distance formula equation, we can substitute the values of the x and y coordinates of A and B:
AB = √((0 - 5)² + (5 - (-5))²)
= √(25 + 100)
= √125
= 5√5
Therefore, the distance between points A and B is 5√5.
Similarly, we can find the distance between points B and C:
BC = √((2 - 0)² + (1 - 5)²)
= √4 + 16
= √20
= 2√5
Finally, we can find the distance between points A and C:
AC = √((2 - 5)² + (1 - (-5))²)
= √9 + 36
= √45
= 3√5
Alternatively, we can use the equation of the line passing through any two of these points and check if the third point lies on that line.
Let's use the points A and B to find the equation of the line passing through them:
y - (-5) = ((5 - (-5)) / (0 - 5))(x - 5)
y + 5 = (10 / (-5))(x - 5)
y + 5 = -2(x - 5)
y + 5 = -2x + 10
y = -2x + 5
Now, let's check if point C lies on this line by substituting its coordinates into the equation:
1 = -2(2) + 5
1 = 1
Since the equation is true, we can conclude that points A, B, and C are collinear. Moreover, since point C lies between points A and B on the line, we can say that C lies on segment AB.
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Complete Question:
Given points A, B, and C. Find AB, BC, and AC. Are A, B, and C collinear? If so, which point lies between the other two? A(5, −5), B(0,5), C(2, 1)
If A={1,2,3}, B= {} show that A is not equal to B
In set theory, two sets are considered equal if they have the same elements. In this case, A is a set containing the elements 1, 2, and 3, while B is an empty set (also known as the null set),
A contains three distinct elements, and B contains none, we can conclude that A and B are not equal, i.e., A is not equal to B.
A ≠ B
Set theory is a branch of mathematics that studies collections of objects, called sets, and the relationships between them. A set is defined as a well-defined collection of distinct objects, which can be anything from numbers and letters to more abstract concepts like functions and geometrical shapes. The set theory provides a foundation for other areas of mathematics, including algebra, topology, and logic.
One of the fundamental concepts of set theory is the notion of membership, which states that an object either belongs to a set or does not. Sets can also be combined through operations such as union, intersection, and complementation, and the relationships between sets can be represented using Venn diagrams.
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You are playing a game with a friend. It costs you $2 to play. If you roll a 1 on a 6-sided die you win $4. If you roll a 2, 3, 4, 5, or 6 you win nothing and lose $2 the cost to play. How much should the player be willing to pay to play this game and not lose money in the long run?
The player should be willing to pay up to $1.33 to play this game and not lose money in the long run.
The expected value is the sum of the products of each possible outcome and its probability. Let's calculate the expected value of the game:
E(X) = (1/6) * $4 + (5/6) * (-$2)
E(X) = $0.67
This means that on average, the player can expect to win $0.67 per game. Since it costs $2 to play, the player should not be willing to pay more than $2 - $0.67 = $1.33 to play the game and not lose money in the long run.
Probability theory is based on axioms, which are basic assumptions about the nature of probability. It is used to quantify uncertainty and to make predictions based on the available information. Probability is expressed as a number between 0 and 1, with 0 meaning an event is impossible, and 1 meaning an event is certain.
The concept of probability is used in a variety of fields, including statistics, economics, engineering, and physics. In statistics, probability is used to model random variables, estimate parameters, and test hypotheses. In economics, probability is used to model financial risks and decision-making under uncertainty. In engineering and physics, probability is used to model complex systems and predict the behavior of particles.
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Two ropes are attached to a tree, and forces of F_1 = 1.31 + 4.6J n and F_2 = 3.2i + 6.8j n are applied. The forces are coplanar (in the same plane). What is the resultant (net force) of these two force vectors (in N)? (Express your answer in vector form.) Find the magnitude (in N) and direction (in degrees counterclockwise from the +x-axis) of this net force.
The magnitude and direction of the net force are found by adding the two forces together as resultant force vectors.
a) 11.82 N
b) 74.07°
To find the net force, we add the two force vectors F_1 and F_2:
Fnet = F_1 + F_2
Fnet = (1.31 + 4.6j) N + (3.2i + 6.8j) N
Fnet = 3.2i + (1.31 + 4.6j + 6.8j) N
Fnet = 3.2i + (1.31 + 11.4j) N
To find the magnitude of the net force, we use the Pythagorean theorem:
|Fnet| = sqrt[(3.2)^2 + (1.31 + 11.4)^2] N
|Fnet| ≈ 11.6 N
To find the direction of the net force, we use the inverse tangent function:
θ = tan^(-1)(y/x)
θ = tan^(-1)(11.4/3.2)
θ ≈ 73.8 degrees
Since the net force is in the first quadrant, the direction counterclockwise from the +x-axis is simply θ:
Direction = 73.8 degrees counterclockwise from the +x-axis
Therefore, the net force is Fnet = 3.2i + (1.31 + 11.4j) N, with a magnitude of approximately 11.6 N and a direction of approximately 73.8 degrees counterclockwise from the +x-axis.
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