Question 1

Given the cost function, C(x), and the revenue function, R(x), find the number of units x that must be sold to break even.
-A vendor sells hot dogs, bags of potato chips, and soft drinks. A customer buys 3 hot dogs, 3 bags of potato chips, and 3 soft drinks for $11.25. The price of a hot dog is $1.25 more than the price of a bag of potato chips.
The cost of a soft drink is $2.00 less than the price of two hot dogs. Find the cost of each item.

Accepted Answer

Let the price of a bag of potato chips be x. Then, the price of a hot dog is x + 1.25, and the price of a soft drink is 2(x + 1.25) - 2. Solving the equation 3(x + 1.25) + 3x + 3(2(x + 1.25) - 2) = 11.25 gives x = 0.50. Thus, the prices are $1.75 for a hot dog, $0.50 for a bag of potato chips, and $1.50 for a soft drink.

Question 2

Given the cost function, C(x), and the revenue function, R(x), find the number of units x that must be sold to break even.
-$$\begin{array} { l } 
C ( x ) = 6000 x + 70,000 \
R ( x ) = 16,000 x
\end{array}$$

Accepted Answer

To find the break-even point, set the cost function equal to the revenue function: $6000x + 70,000 = 16,000x$. Solving for $x$ gives $x = 7$, meaning 7 units must be sold to break even.

Question 3

Given the cost function, C(x), and the revenue function, R(x), find the number of units x that must be sold to break even.
-$$\begin{array} { l } 
C ( x ) = 11 x + 15,200 \
R ( x ) = 30 x
\end{array}$$

Accepted Answer

The answer of Given the cost function, C(x), and the...

Question 4

Solve.
-University Theater sold 562 tickets for a play. Tickets cost $23 per adult and $13 per senior citizen. If total receipts were $9086, how many senior citizen tickets were sold?

Accepted Answer

The answer of Solve.
-University Theater sold 562 tickets for a...

Question 5

Solve.
-The manager of a bulk foods establishment sells a trail mix for $5 per pound and premium cashews for $11 per pound. The manager wishes to make a 120-pound trail mix-cashew mixture that will sell for $9 per pound.
How many pounds of each should be used?

Accepted Answer

The answer of Solve.
-The manager of a bulk foods establishment...

Question 6

Given the cost function, C(x), and the revenue function, R(x), find the number of units x that must be sold to break even.
-$$\begin{array} { l } 
C ( x ) = 88 x + 2800 \
R ( x ) = 108 x
\end{array}$$

Accepted Answer

To find the break-even point, set the cost function equal to the revenue function: $88x + 2800 = 108x$. Solving for $x$, we get $2800 = 20x$, so $x = 140$ units.

Question 7

Given the cost function, C(x), and the revenue function, R(x), find the number of units x that must be sold to break even.
-Find the values of $a , b$, and c such that the equation $y = a x ^ { 2 } + b x + c$ has ordered pair solutions $( - 2 , - 6 )$, $( 2,10 )$, and $( 4,6 )$.

Accepted Answer

To find the values of $a$, $b$, and $c$, we can substitute the given points into the equation $y = ax^2 + bx + c$ and solve the resulting system of equations. Substituting $(-2, -6)$, we get $-6 = 4a - 2b + c$. Substituting $(2, 10)$, we get $10 = 4a + 2b + c$. Substituting $(4, 6)$, we get $6 = 16a + 4b + c$. Solving this system of equations yields $a = -1$, $b = 4$, and $c = 6$.

Question 8

Solve.
-A chemist needs 170 milliliters of a 59% solution but has only 15% and 83% solutions available. Find how many milliliters of each that should be mixed to get the desired solution.

Accepted Answer

To find the correct amounts, set up the equation: 0.15x + 0.83(170-x) = 0.59(170). Solving this gives x = 60 ml of 15% solution and 110 ml of 83% solution.

Question 9

Given the cost function, C(x), and the revenue function, R(x), find the number of units x that must be sold to break even.
-At the break-even point both cost and revenue are what?

Accepted Answer

At the break-even point, the cost and revenue are equal. Therefore, the number of units x that must be sold to break even is when C(x) = R(x). The given choice D, $2250, represents the point where cost and revenue are equal.

Question 10

Given the cost function, C(x), and the revenue function, R(x), find the number of units x that must be sold to break even.
-$$\begin{array} { l } 
C ( x ) = 0.3 x + 1320 \
R ( x ) = 1.5 x
\end{array}$$

Accepted Answer

To find the break-even point, set the cost function equal to the revenue function: $0.3x + 1320 = 1.5x$. Solving for $x$, subtract $0.3x$ from both sides to get $1320 = 1.2x$, then divide both sides by $1.2$ to find $x = 1100$.

Question 11

Solve.
-A certain aircraft can fly 1190 miles with the wind in 5 hours and travel the same distance against the wind in 7 hours. What is the speed of the wind?

Accepted Answer

The speed of the aircraft in still air is 238 mph (1190 miles / 5 hours) and 170 mph against the wind (1190 miles / 7 hours). The wind speed is the difference divided by 2, which is 34 mph.

Question 12

Solve.
-A vendor sells hot dogs and bags of potato chips. A customer buys 5 hot dogs and 3 bags of potato chips for $9.25. Another customer buys 4 hot dogs and 5 bags of potato chips for $10.00. Find the cost of each item.

Accepted Answer

The answer of Solve.
-A vendor sells hot dogs and bags...

Question 13

Given the cost function, C(x), and the revenue function, R(x), find the number of units x that must be sold to break even.
-A basketball player scored 16 points in a game. The number of three-point field goals the player made was 14 less than three times the number of free throws (each worth 1 point). Twice the number of two-point field goals
The player made was 7 more than the number of three-point field goals made. Find the number of free-throws,
Two-point field goals, and three-point field goals that the player made in the game.

Accepted Answer

Let F, T, and Th represent the number of free throws, two-point field goals, and three-point field goals, respectively. The total points can be represented as F + 2T + 3Th = 16. The problem states Th = 3F - 14 and 2T = Th + 7. Solving these equations simultaneously, we find F = 5, T = 4, and Th = 1, which corresponds to choice B.

Question 14

Given the cost function, C(x), and the revenue function, R(x), find the number of units x that must be sold to break even.
-A store sells tents, sleeping bags, and camp stools. A customer buys a tent, 4 sleeping bags, and 3 camp stools for $220. The price of the tent is 7 times the cost of a camp stool. The cost of a sleeping bag is $20 more than the
Cost of a camp stool. Find the cost of each item.

Accepted Answer

Let the cost of a camp stool be $x. Then, the cost of a tent is 7x, and the cost of a sleeping bag is x + $20. The total cost for a customer buying a tent, 4 sleeping bags, and 3 camp stools is given by: 7x + 4(x + 20) + 3x = $220. Solving this equation: 14x + 80 = 220, 14x = 140, x = $10. Therefore, the cost of a tent is $70 (7x), the cost of a sleeping bag is $30 (x + 20), and the cost of a camp stool is $10.

Question 15

Given the cost function, C(x), and the revenue function, R(x), find the number of units x that must be sold to break even.
-Three trains  one eastbound, one westbound, and one northbound  leave a city at the same time. The speed of the northbound train is 10 miles per hour greater than the speed of the eastbound train. After 2 hours, the
Distance between the westbound train and the eastbound train is 200 miles. Twice the speed of the westbound
Train is 100 miles per hour more than the speed of the northbound train. Find the speeds of the three trains.

Accepted Answer

To solve this, we set up equations based on the information given. Let $E$ be the speed of the eastbound train, $W$ be the speed of the westbound train, and $N$ be the speed of the northbound train. We know that $N = E + 10$. The distance between the westbound and eastbound trains after 2 hours is 200 miles, so $2E + 2W = 200$. Finally, twice the speed of the westbound train is 100 mph more than the speed of the northbound train, so $2W = N + 100$. Substituting $N = E + 10$ into the last equation and solving the system of equations gives us $E = 30$ mph, $W = 70$ mph, and $N = 40$ mph.

Question 16

Given the cost function, C(x), and the revenue function, R(x), find the number of units x that must be sold to break even.
-Use the revenue and cost functions to write the profit function from producing and selling $x$ binoculars.

Accepted Answer

The profit function, $P(x)$, is defined as the revenue function, $R(x)$, minus the cost function, $C(x)$. Therefore, if $R(x) = a\cdot x$ and $C(x) = b + a\cdot x$, where $b$ is the fixed cost, then $P(x) = R(x) - C(x) = a\cdot x - (b + a\cdot x) = a\cdot x - b$. The correct option, $P(x) = 2x - 1500$, reflects this relationship, indicating a linear profit function where the profit increases by 2 units for every additional unit sold, after accounting for a fixed cost of 1500.

Question 17

Given the cost function, C(x), and the revenue function, R(x), find the number of units x that must be sold to break even.
-$$\begin{array} { l } 
C ( x ) = 143 x + 306,000 \
R ( x ) = 313 x
\end{array}$$

Accepted Answer

To find the break-even point, set the cost function equal to the revenue function: $143x + 306,000 = 313x$. Solving for $x$ gives $x = 1800$, meaning 1800 units must be sold to break even.

Question 18

Given the cost function, C(x), and the revenue function, R(x), find the number of units x that must be sold to break even.
-What is the profit when 796 binoculars are produced?

Accepted Answer

The answer of Given the cost function, C(x), and the...

Question 19

Solve.
-Two cars leave a city and head in the same direction. After 5 hours, the faster car is 15 miles ahead of the slower car. The slower car has traveled 230 miles. Find the speeds of the two cars.

Accepted Answer

The slower car's speed is 230 miles / 5 hours = 46 mph. The faster car is 15 miles ahead, having traveled 245 miles in the same time, so its speed is 245 miles / 5 hours = 49 mph.

Question 20

Solve the system of linear equations using matrices.
-$$\left\{ \begin{array} { l } 
x + y = - 6 \
x - y = 2
\end{array} \right.$$

Accepted Answer

To solve the system of equations using matrices, we can represent the system in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix. For the given system:$$A = \begin{pmatrix} 1 & 1 \ 1 & -1 \end{pmatrix}, X = \begin{pmatrix} x \ y \end{pmatrix}, B = \begin{pmatrix} -6 \ 2 \end{pmatrix}$$To find X, we calculate $A^{-1}B$:$$A^{-1} = \frac{1}{(1)(-1) - (1)(1)} \begin{pmatrix} -1 & -1 \ -1 & 1 \end{pmatrix} = \frac{1}{-2} \begin{pmatrix} -1 & -1 \ -1 & 1 \end{pmatrix} = \begin{pmatrix} 0.5 & 0.5 \ 0.5 & -0.5 \end{pmatrix}$$$$X = A^{-1}B = \begin{pmatrix} 0.5 & 0.5 \ 0.5 & -0.5 \end{pmatrix} \begin{pmatrix} -6 \ 2 \end{pmatrix} = \begin{pmatrix} -2 \ -4 \end{pmatrix}$$Therefore, the solution to the system of equations is $x = -2, y = -4$, which corresponds to choice A $(-2, -4)$.

Quiz 4: Systems of Equations and Inequalities

Given the cost function, C(x) , and the revenue function, R(x) , find the number of units x that must be sold to break even. - $\begin{array} { l } C ( x ) = 6000 x + 70,000 \\R ( x ) = 16,000 x\end{array}$

Given the cost function, C(x) , and the revenue function, R(x) , find the number of units x that must be sold to break even. - $\begin{array} { l } C ( x ) = 11 x + 15,200 \\R ( x ) = 30 x\end{array}$

Solve. -University Theater sold 562 tickets for a play. Tickets cost $23 per adult and $13 per senior citizen. If total receipts were $9086, how many senior citizen tickets were sold?

Solve. -The manager of a bulk foods establishment sells a trail mix for $5 per pound and premium cashews for $11 per pound. The manager wishes to make a 120-pound trail mix-cashew mixture that will sell for $9 per pound. How many pounds of each should be used?

Given the cost function, C(x) , and the revenue function, R(x) , find the number of units x that must be sold to break even. - $\begin{array} { l } C ( x ) = 88 x + 2800 \\R ( x ) = 108 x\end{array}$

Given the cost function, C(x) , and the revenue function, R(x) , find the number of units x that must be sold to break even. -Find the values of $a , b$ , and c such that the equation $y = a x ^ { 2 } + b x + c$ has ordered pair solutions $( - 2 , - 6 )$ , $( 2,10 )$ , and $( 4,6 )$ .

Solve. -A chemist needs 170 milliliters of a 59% solution but has only 15% and 83% solutions available. Find how many milliliters of each that should be mixed to get the desired solution.

Given the cost function, C(x) , and the revenue function, R(x) , find the number of units x that must be sold to break even. -At the break-even point both cost and revenue are what?

Given the cost function, C(x) , and the revenue function, R(x) , find the number of units x that must be sold to break even. - $\begin{array} { l } C ( x ) = 0.3 x + 1320 \\R ( x ) = 1.5 x\end{array}$

Solve. -A certain aircraft can fly 1190 miles with the wind in 5 hours and travel the same distance against the wind in 7 hours. What is the speed of the wind?

Solve. -A vendor sells hot dogs and bags of potato chips. A customer buys 5 hot dogs and 3 bags of potato chips for $9.25. Another customer buys 4 hot dogs and 5 bags of potato chips for $10.00. Find the cost of each item.

Given the cost function, C(x) , and the revenue function, R(x) , find the number of units x that must be sold to break even. -Use the revenue and cost functions to write the profit function from producing and selling $x$ binoculars.

Given the cost function, C(x) , and the revenue function, R(x) , find the number of units x that must be sold to break even. - $\begin{array} { l } C ( x ) = 143 x + 306,000 \\R ( x ) = 313 x\end{array}$

Given the cost function, C(x) , and the revenue function, R(x) , find the number of units x that must be sold to break even. -What is the profit when 796 binoculars are produced?

Solve. -Two cars leave a city and head in the same direction. After 5 hours, the faster car is 15 miles ahead of the slower car. The slower car has traveled 230 miles. Find the speeds of the two cars.

Solve the system of linear equations using matrices. - $\left\{ \begin{array} { l } x + y = - 6 \\x - y = 2\end{array} \right.$

Quiz 4: Systems of Equations and Inequalities

Given the cost function, C(x) , and the revenue function, R(x) , find the number of units x that must be sold to break even. - C(x) =6000x+70,000R(x) =16,000x\begin{array} { l } C ( x ) = 6000 x + 70,000 \\R ( x ) = 16,000 x\end{array}C(x) =6000x+70,000R(x) =16,000x​

Given the cost function, C(x) , and the revenue function, R(x) , find the number of units x that must be sold to break even. - C(x) =11x+15,200R(x) =30x\begin{array} { l } C ( x ) = 11 x + 15,200 \\R ( x ) = 30 x\end{array}C(x) =11x+15,200R(x) =30x​