Question 1

Find the dimensions of the rectangle meeting the specified condition. ​
The perimeter is 70 inches and the width is three-fourths the length.
​
A)22 in × 15 in
B)24 in × 19 in
C)22 in × 17 in
D)20 in × 19 in
E)20 in × 15 in

Accepted Answer

The answer of Find the dimensions of the rectangle meeting...

Question 2

Find the sales necessary to break even (R = C) for the cost C of producing x units and the revenue R obtained by selling x units.(Round to the nearest whole unit.)  $$C = 6.5 \sqrt { x } + 9,674 , R = 3.2 x$$  
A)2,937 units
B)3,137 units
C)3,037 units
D)3,337 units
E)3,237 units

Accepted Answer

The answer of Find the sales necessary to break even...

Question 3

Use a graphing utility to solve the system of equations.Find the solution(s) accurate to two decimal places.  $$\left\{ \begin{aligned}
y & = e ^ { x + 4 } \
x - y + 5 & = 0
\end{aligned} \right.$$  
A)   (-4, 1)
B)   (0, 4)
C)  (0, -4)
D)   (4, 4)
E)   (-4, 1)

Accepted Answer

The answer of Use a graphing utility to solve the...

Question 4

What are the dimensions of a rectangular tract of land if its perimeter is 50 kilometers and its area is 150 square kilometers? ​
A)14 km × 19 km
B)10 km × 19 km
C)12 km × 15 km
D)10 km × 15 km
E)12 km × 17 km

Accepted Answer

The dimensions that satisfy both the perimeter and area conditions are 10 km × 15 km. The perimeter of a rectangle is given by 2(l + w) = 50 km, and the area is l × w = 150 sq km. Only the dimensions 10 km and 15 km satisfy both equations: 2(10 + 15) = 50 km for the perimeter and 10 × 15 = 150 sq km for the area.

Question 5

Solve the system graphically.  $\left\{ \begin{aligned}
6 x + 8 y & = 48 \
x - 8 y & = - 20
\end{aligned} \right.$  
A)(6, 3)
B)(-4, -3)
C)(4, 3)
D)(6, 6)
E)(-6, -6)

Accepted Answer

To solve the system graphically, we first convert each equation into slope-intercept form (y = mx + b). For the first equation, $6x + 8y = 48$, we get $y = -\frac{3}{4}x + 6$. For the second equation, $x - 8y = -20$, we get $y = \frac{1}{8}x - \frac{5}{2}$. Plotting these two lines on a graph, we find that they intersect at the point (4, 3), which corresponds to option C.

Question 6

Solve the system graphically.  $\left\{ \begin{aligned}
x - y & = 1 \
6 x - 3 y & = 24
\end{aligned} \right.$  
A)(-8, -7)
B)(-7, -6)
C)(7, 6)
D)(7, 7)
E)(8, 7)

Accepted Answer

To solve the system graphically, we first rewrite each equation in slope-intercept form (y = mx + b). For the first equation, $x - y = 1$, we get $y = x - 1$. For the second equation, $6x - 3y = 24$, dividing everything by 3 gives $2x - y = 8$, or $y = 2x - 8$. Graphing these two lines, we find that they intersect at the point (7, 6), which is option C.

Question 7

A small software company invests $24,425 to produce a software package that will sell for $70.Each unit can be produced for $45.84.How many units must be sold to break even? ​
A)911 units
B)811 units
C)1211 units
D)1011 units
E)1111 units

Accepted Answer

To find the break-even point, we need to calculate the total cost (investment) and the total revenue (sales).

Total cost = fixed cost + variable cost

The fixed cost in this scenario is the initial investment of $24,425.

The variable cost is the cost of producing each unit, which is $45.84.

So, the total cost per unit is:

Total cost per unit = $24,425 + $45.84 = $24,470.84

To find the break-even point, we need to divide the total cost by the profit per unit:

Break-even point = total cost / profit per unit

Profit per unit = sales price - cost per unit

Profit per unit = $70 - $45.84 = $24.16

Break-even point = $24,470.84 / $24.16 ≈ 1011 units

Therefore, the company must sell 1011 units to break even. The answer is D) 1011 units.

Question 8

Find the dimensions of the rectangle meeting the specified condition.  
The perimeter is 528 feet and the length is $4 \frac { 1 } { 2 }$ times the width.
 
A)216 ft × 52 ft
B)216 ft × 48 ft
C)220 ft × 52 ft
D)218 ft × 48 ft
E)218 ft × 50 ft

Accepted Answer

Let the width be $ w $. Then the length is $ 4.5w $. The perimeter is given by $ 2(w + 4.5w) = 528 $. Solving for $ w $, we get $ w = 48 $ ft and the length $ 4.5 \times 48 = 216 $ ft. Thus, the dimensions are 216 ft × 48 ft.

Question 9

Solve the system graphically.  $\left\{ \begin{array} { l } 
x - y = 0 \
5 x - 2 y = 9
\end{array} \right.$
A)(5, 5)
B)(3, 3)
C)(-5, -5)
D)(-3, -3)
E)(5, 3)

Accepted Answer

To solve the system graphically, we first rewrite each equation in slope-intercept form (y = mx + b). For the first equation, x - y = 0 becomes y = x. For the second equation, 5x - 2y = 9 becomes y = (5/2)x - (9/2). Graphing these two lines, we find that they intersect at the point (3, 3), which is option B.

Question 10

Find the dimensions of the rectangle meeting the specified condition. ​
The perimeter is 352 centimeters and the length is 4 centimeters greater than the width.
​
A)90 cm × 90 cm
B)90 cm × 86 cm
C)92 cm × 88 cm
D)94 cm × 90 cm
E)92 cm × 86 cm

Accepted Answer

The perimeter of a rectangle is given by the formula $2l + 2w$, where $l$ is the length and $w$ is the width. Given that the length is 4 cm greater than the width ($l = w + 4$), and the perimeter is 352 cm, we can set up the equation $2(w + 4) + 2w = 352$. Simplifying gives $4w + 8 = 352$, and further simplification gives $w = 86$ cm. Therefore, the length is $86 + 4 = 90$ cm, making the dimensions $90$ cm by $86$ cm.

Question 11

The total weekly sales for a newly released portable media player (PMP) increased each week.At the same time, the total weekly sales for another newly released PMP decreased each week.Models that approximate the total weekly sales S (in thousands of units) for each PMP are  $$\left\{ \begin{array} { l l } 
S = 10 x + 70 & \text { PMP1 } \
S = - 10 x + 210 & \text { PMP2 }
\end{array} \right.$$   where x represents the number of weeks each PMP was in stores, with x = 0 corresponding to the PMP sales on the day each PMP was first released in stores.After how many weeks will the sales for the two PMPs be equal 
 
A)11 weeks
B)9 weeks
C)7 weeks
D)3 weeks
E)5 weeks

Accepted Answer

To find when the sales are equal, set the equations equal: 10x + 70 = -10x + 210. Solving for x gives x = 7 weeks.

Question 12

Solve the system by the method of substitution. $\left\{ \begin{array} { l } 
0.5 x + 3.2 y = 5.3 \
0.3 x - 1.4 y = - 1.8
\end{array} \right.$  
A)(3, 3)
B) $\left( - 1 , \frac { 3 } { 2 } \right)$
C) $\left( 1 , \frac { 3 } { 2 } \right)$
D) $\left( - 1 , - \frac { 3 } { 2 } \right)$
E)(-3, -3)

Accepted Answer

First, solve one of the equations for $x$ or $y$. Let's solve the first equation for $x$: $0.5x = 5.3 - 3.2y$, so $x = 10.6 - 6.4y$. Substitute this expression for $x$ in the second equation: $0.3(10.6 - 6.4y) - 1.4y = -1.8$. Simplify and solve for $y$, then substitute $y$ back into one of the original equations to find $x$. Solving gives $y = \frac{3}{2}$ and $x = 1$, so the correct answer is $\left( 1 , \frac { 3 } { 2 } \right)$.

Question 13

The weekly rentals for a newly released DVD of an animated film at a local video store decreased each week.At the same time, the weekly rentals for a newly released DVD of a horror film increased each week.Models that approximate the weekly rentals R for each DVD are
 $$\left\{ \begin{array} { l l } 
R = 380 - 24 x & \text { Animated film } \
R = 28 + 20 x & \text { Horror film }
\end{array} \right.$$   where x represents the number of weeks each DVD was in the store, with x = 1 corresponding to the first week.After how many weeks will the rentals for the two movies be equal 
 
A)10 weeks
B)6 weeks
C)4 weeks
D)12 weeks
E)8 weeks

Accepted Answer

To find when the rentals for the two movies will be equal, set the two equations equal to each other and solve for $x$:$$380 - 24x = 28 + 20x$$Combining like terms:$$352 = 44x$$Solving for $x$:$$x = 8$$Therefore, after 8 weeks, the rentals for the two movies will be equal.

Question 14

What are the dimensions of an isosceles right triangle with a four-inch hypotenuse and an area of 4 square inches 
A) $2 \sqrt { 2 }$ in. × $2 \sqrt { 2 }$ in. × 4 in.
B) $\sqrt { 10 }$ in. × $\sqrt { 10 }$ in. × 4 in.
C) $3$ in. × $3$ in. × 4 in.
D) $\sqrt { 11 }$ in. × $\sqrt { 11 }$ in. × 4 in.
E) $2 \sqrt { 3 }$ in. × $2 \sqrt { 3 }$ in. × 4 in.

Accepted Answer

In an isosceles right triangle, the legs are equal, and the hypotenuse is $leg \times \sqrt{2}$. Given a 4-inch hypotenuse, each leg is $4/\sqrt{2} = 2\sqrt{2}$ inches, matching option A.

Question 15

Solve the system graphically or algebraically.Find the solution(s) accurate to three decimal places.  $\left\{ \begin{array} { r } 
y - e ^ { - x } = 3 \
y - \ln x = 5
\end{array} \right.$  
A)(3.752, 0.287)
B)(0.387, 3.852)
C)(0.287, 3.752)
D)(0.287, 3.852)
E)(3.852, 0.387)

Accepted Answer

To solve the system algebraically, we can set the two equations equal to each other since they both equal $y$:$$y = e^{-x} + 3$$$$y = \ln(x) + 5$$Setting them equal to each other gives:$$e^{-x} + 3 = \ln(x) + 5$$Solving this equation requires numerical methods or graphing because it involves both $e^{-x}$ and $\ln(x)$. Graphing these equations or using a numerical solver, we find that the solution is approximately $x = 0.287, y = 3.752$, which corresponds to choice C.

Question 16

Find the sales necessary to break even (R = C) for the cost C of producing x units and the revenue R obtained by selling x units.(Round to the nearest whole unit.) ​
C = 8630x + 264,000, R = 9950x
​
A)100 units
B)300 units
C)0 unit
D)400 units
E)200 units

Accepted Answer

To find the break-even point, set the cost equal to the revenue: $8630x + 264,000 = 9950x$. Solving for $x$, we get $x = 200$.

Question 17

Solve the system graphically.  $\left\{ \begin{aligned}
x ^ { 2 } + y ^ { 2 } & = 85 \
( x - 5 ) ^ { 2 } + y ^ { 2 } & = 50
\end{aligned} \right.$  
A)(8, -8), (8, 8)
B)(-8, -8), (-8, 8)
C)(-6, -7), (6, 7)
D)(8, 7), (8, 8)
E)(6, 7), (6, -7)

Accepted Answer

The first equation represents a circle centered at the origin (0,0) with a radius of $\sqrt{85}$. The second equation represents a circle centered at (5,0) with a radius of $\sqrt{50}$. The correct intersections, based on the graphical solution, are (6, 7) and (6, -7).

Question 18

Solve the system graphically.  $\left\{ \begin{array} { r } 
- x + 2 y = 3 \
x - y = 2
\end{array} \right.$  
A)(-7, -5)
B)(9, -9)
C)(7, 5)
D)(9, 5)
E)(-9, -9)

Accepted Answer

To solve the system graphically, we first convert each equation into slope-intercept form (y = mx + b). For the first equation, -x + 2y = 3, we add x to both sides and then divide by 2 to get y = 0.5x + 1.5. For the second equation, x - y = 2, we subtract x from both sides and multiply by -1 to get y = x - 2. Plotting these two lines on a graph, we find that they intersect at the point (7, 5), which is option C.

Question 19

Find the dimensions of the rectangle meeting the specified condition. ​
The perimeter is 36 meters and the length is 4 meters greater than the width.
​
A)13 m × 7 m
B)13 m × 9 m
C)11 m × 7 m
D)15 m × 11 m
E)11 m × 11 m

Accepted Answer

Let's say the width is x meters. Then the length is 4 meters more than that, so it's x + 4 meters.

The perimeter formula for a rectangle is: P = 2L + 2W

If we substitute in our values, we get:

36 = 2(x + 4) + 2x

Simplifying, we get:

36 = 4x + 8

28 = 4x

x = 7

So the width is 7 meters, and the length is 4 meters more than that, so it's 11 meters.

Therefore, the dimensions of the rectangle are 11 meters by 7 meters.

The correct answer choice is C.

Question 20

A small fast-food restaurant invests $5,951 to produce a new food item that will sell for $3.81.Each unit can be produced for $2.50.How many items must be sold to break even? ​
A)4543 items
B)4743 items
C)4343 items
D)4643 items
E)4443 items

Accepted Answer

The answer of A small fast-food restaurant invests $5,951 to...

Quiz 7: Systems Of Equations and Inequalities

Find the dimensions of the rectangle meeting the specified condition. The perimeter is 70 inches and the width is three-fourths the length.

Find the sales necessary to break even (R = C) for the cost C of producing x units and the revenue R obtained by selling x units.(Round to the nearest whole unit.) $C = 6.5 \sqrt { x } + 9,674 , R = 3.2 x$

Use a graphing utility to solve the system of equations.Find the solution(s) accurate to two decimal places. $\left\{ \begin{aligned}y & = e ^ { x + 4 } \\x - y + 5 & = 0\end{aligned} \right.$

What are the dimensions of a rectangular tract of land if its perimeter is 50 kilometers and its area is 150 square kilometers?

Solve the system graphically. $\left\{ \begin{aligned}6 x + 8 y & = 48 \\x - 8 y & = - 20\end{aligned} \right.$

Solve the system graphically. $\left\{ \begin{aligned}x - y & = 1 \\6 x - 3 y & = 24\end{aligned} \right.$

A small software company invests $24,425 to produce a software package that will sell for $70.Each unit can be produced for $45.84.How many units must be sold to break even?

Find the dimensions of the rectangle meeting the specified condition. The perimeter is 528 feet and the length is $4 \frac { 1 } { 2 }$ times the width.

Solve the system graphically. $\left\{ \begin{array} { l } x - y = 0 \\5 x - 2 y = 9\end{array} \right.$

Find the dimensions of the rectangle meeting the specified condition. The perimeter is 352 centimeters and the length is 4 centimeters greater than the width.

Solve the system by the method of substitution. $\left\{ \begin{array} { l } 0.5 x + 3.2 y = 5.3 \\0.3 x - 1.4 y = - 1.8\end{array} \right.$

What are the dimensions of an isosceles right triangle with a four-inch hypotenuse and an area of 4 square inches

Solve the system graphically or algebraically.Find the solution(s) accurate to three decimal places. $\left\{ \begin{array} { r } y - e ^ { - x } = 3 \\y - \ln x = 5\end{array} \right.$

Find the sales necessary to break even (R = C) for the cost C of producing x units and the revenue R obtained by selling x units.(Round to the nearest whole unit.) C = 8630x + 264,000, R = 9950x

Solve the system graphically. $\left\{ \begin{aligned}x ^ { 2 } + y ^ { 2 } & = 85 \\( x - 5 ) ^ { 2 } + y ^ { 2 } & = 50\end{aligned} \right.$

Solve the system graphically. $\left\{ \begin{array} { r } - x + 2 y = 3 \\x - y = 2\end{array} \right.$

Find the dimensions of the rectangle meeting the specified condition. The perimeter is 36 meters and the length is 4 meters greater than the width.

A small fast-food restaurant invests $5,951 to produce a new food item that will sell for $3.81.Each unit can be produced for $2.50.How many items must be sold to break even?

Quiz 7: Systems Of Equations and Inequalities

Find the dimensions of the rectangle meeting the specified condition. ​ The perimeter is 70 inches and the width is three-fourths the length. ​

Find the sales necessary to break even (R = C) for the cost C of producing x units and the revenue R obtained by selling x units.(Round to the nearest whole unit.) C=6.5x+9,674,R=3.2xC = 6.5 \sqrt { x } + 9,674 , R = 3.2 xC=6.5x​+9,674,R=3.2x

Use a graphing utility to solve the system of equations.Find the solution(s) accurate to two decimal places. {y=ex+4x−y+5=0\left\{ \begin{aligned}y & = e ^ { x + 4 } \\x - y + 5 & = 0\end{aligned} \right.{yx−y+5​=ex+4=0​

What are the dimensions of a rectangular tract of land if its perimeter is 50 kilometers and its area is 150 square kilometers? ​

Solve the system graphically. {6x+8y=48x−8y=−20\left\{ \begin{aligned}6 x + 8 y & = 48 \\x - 8 y & = - 20\end{aligned} \right.{6x+8yx−8y​=48=−20​

Solve the system graphically. {x−y=16x−3y=24\left\{ \begin{aligned}x - y & = 1 \\6 x - 3 y & = 24\end{aligned} \right.{x−y6x−3y​=1=24​

A small software company invests $24,425 to produce a software package that will sell for $70.Each unit can be produced for $45.84.How many units must be sold to break even? ​

Find the dimensions of the rectangle meeting the specified condition. The perimeter is 528 feet and the length is 4124 \frac { 1 } { 2 }421​ times the width.

Solve the system graphically. {x−y=05x−2y=9\left\{ \begin{array} { l } x - y = 0 \\5 x - 2 y = 9\end{array} \right.{x−y=05x−2y=9​

Find the dimensions of the rectangle meeting the specified condition. ​ The perimeter is 352 centimeters and the length is 4 centimeters greater than the width. ​

Solve the system by the method of substitution. {0.5x+3.2y=5.30.3x−1.4y=−1.8\left\{ \begin{array} { l } 0.5 x + 3.2 y = 5.3 \\0.3 x - 1.4 y = - 1.8\end{array} \right.{0.5x+3.2y=5.30.3x−1.4y=−1.8​