Question 1

A rug is to fit in a room so that a border of even width is left on all four sides. If the room is 17 feet by 22 feet and the area of the rug is 204 square feet, how wide will the border be?
A) 5 feet
B) 2.5 feet
C) 4.5 feet
D) 3.5 feet

Accepted Answer

The area of the room is 17 * 22 = 374 square feet. The area of the rug is 204 square feet, so the area of the border is 374 - 204 = 170 square feet. Let x be the width of the border. The dimensions of the rug will then be (17 - 2x) by (22 - 2x). The area of the rug is also given by the equation (17 - 2x)(22 - 2x) = 204. Simplifying this equation gives a quadratic equation in terms of x. Solving for x, we find that x = 2.5 feet.

Question 2

A rectangular garden has dimensions of 18 feet by 15 feet. A gravel path of equal width is to be built around the garden. How wide can the path be if there is enough gravel for 234 square feet?
A) 5.5 feet
B) 5 feet
C) 4 feet
D) 3 feet

Accepted Answer

The area of the garden is 18 feet * 15 feet = 270 square feet. The total area, including the path, minus the garden area equals the area available for the path, which is 234 square feet. Let the width of the path be x feet. The dimensions of the garden plus the path will be (18 + 2x) by (15 + 2x). The equation for the area of the path is: (18 + 2x)(15 + 2x) - 270 = 234. Simplifying this equation to find x is complex, so instead, we can test the options given:For D) 3 feet: The dimensions including the path would be 24 feet by 21 feet. The total area would be 24 * 21 = 504 square feet. Subtracting the garden's area (270 square feet) gives us 234 square feet for the path, which matches the gravel available. Therefore, the path can be 3 feet wide.

Question 3

The area of a square lawn is numerically 12 feet more than the perimeter. Find the length of the side.
A) 72 feet
B) 24 feet
C) 6 feet
D) 18 feet

Accepted Answer

The area of a square is given by $s^2$ and the perimeter is given by $4s$, where $s$ is the side length. The problem states that the area is numerically 12 feet more than the perimeter, so we have the equation $s^2 = 4s + 12$. Rearranging, we get $s^2 - 4s - 12 = 0$. Factoring or using the quadratic formula, we find that $s = 6$ is the positive solution that makes sense in this context, as side lengths cannot be negative.

Question 4

A rectangular space of 390 square feet is allocated for the living and dining areas in an apartment. Find the width of the square living area given that the width of the dining area is 11 feet.

A) 14 feet
B) 26 feet
C) 17 feet
D) 15 feet

Accepted Answer

The answer of A rectangular space of 390 square feet...

Question 5

The height of a triangle is 4 centimeters more than the base. If the area of the triangle is 126 square centimeters, find the base and height of the triangle.
A) Base: 10 centimeters; Height: 14 centimeters
B) Base: 13 centimeters; Height: 19 centimeters
C) Base: 14 centimeters; Height: 18 centimeters
D) Base: 13 centimeters; Height: 17 centimeters

Accepted Answer

The answer of The height of a triangle is 4...

Question 6

A triangle is 6 centimeters wider than it is tall. The area is 140 square $\mathrm{cm}$. Find the width (length of the base).

A) 20 centimeters
B) 6 centimeters
C) 21 centimeters
D) 14 centimeters

Accepted Answer

The answer of A triangle is 6 centimeters wider than...

Question 7

If a ball is thrown upward from the ground with an initial velocity of 48 feet per second, its height after $t$ seconds is given by the function $h(t)=-16 t^{2}+48 t$. Find the number of seconds before the ball lands on the ground.
A) 24 seconds
B) 3 seconds
C) 1.5 seconds
D) 32 seconds

Accepted Answer

The ball lands on the ground when $h(t) = 0$. Solving the equation $-16t^2 + 48t = 0$ for $t$, we factor out $16t$, getting $16t(-t + 3) = 0$. This gives us $t = 0$ (when the ball is thrown) and $t = 3$ seconds (when the ball lands). Therefore, the correct answer is 3 seconds.

Question 8

If a projectile is launched upward from the ground with an initial velocity of 64.5 feet per second, its height $h$ in feet $t$ seconds later is given by the function $h(t)=-16 t^{2}+64.5 t$. After how many seconds does the projectile land on the ground?
A) 4.0 seconds
B) 2.0 seconds
C) 8.1 seconds
D) 0.2 seconds

Accepted Answer

To find when the projectile lands, set $h(t) = 0$ and solve for $t$: $-16t^2 + 64.5t = 0$. Factoring out $t$, we get $t(-16t + 64.5) = 0$, which gives $t = 0$ (launch time) and $t = 4.0$ seconds (landing time).

Question 9

If a projectile is launched upward from a height of 32 feet at an initial velocity of 80 feet per second, then its height after $t$ seconds is given by the function $h(t)=-16 t^{2}+80 t+32$, where $h$ is in feet. After how many seconds will the projectile reach a height of 132 feet?
A) 5 seconds
B) 10 seconds
C) 2.5 seconds
D) 1.25 seconds

Accepted Answer

To find when the projectile reaches a height of 132 feet, set $h(t) = 132$ and solve the equation $-16t^2 + 80t + 32 = 132$. Simplifying gives $-16t^2 + 80t - 100 = 0$. Solving this quadratic equation using the quadratic formula or factoring gives $t = 2.5$ seconds.

Question 10

If a projectile is launched upward from a height of 96 feet at an initial velocity of 80 feet per second, then its height $h$ after $t$ seconds is given by the function $h(t)=-16 t^{2}+80 t+96$. After how many seconds does the projectile land on the ground?
A) 6 seconds
B) 11 seconds
C) 3.0 seconds
D) 5.0 seoonds

Accepted Answer

To find when the projectile lands on the ground, we set $h(t) = 0$ and solve for $t$. Substituting the given function, we get $0 = -16t^2 + 80t + 96$. Solving this quadratic equation, we find that $t = 6$ seconds is the positive solution that represents the time when the projectile lands on the ground.

Question 11

A projectile is launched upward from an initial height of 400 feet, and it lands on the ground after 8 seconds. What was the initial velocity of the projectile?
A) 78 feet/second
B) 88 feet/second
C) 68 feet/second
D) 70 feet/second

Accepted Answer

The correct answer is A because the initial velocity of the projectile is 78 feet/second. This can be calculated using the formula v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration due to gravity (-32 feet/second^2), and t is the time. In this case, the final velocity is 0 feet/second, the time is 8 seconds, and the acceleration due to gravity is -32 feet/second^2. Substituting these values into the formula, we get 0 = u + (-32) * 8, which simplifies to u = 78 feet/second.

Question 12

The height, in feet, of a free-falling object t seconds after being dropped from an initial height s can be found using the function $h(t)=-16 t^{2}+s$. An object is dropped from a helicopter 576 feet above the ground. How long will it take the object to land on the ground?
A) 7.5 seconds
B) 8 seconds
C) 5 seconds
D) 6 seconds

Accepted Answer

To find the time it takes for the object to hit the ground, set $h(t) = 0$ and solve for $t$. Using the given function $h(t) = -16t^2 + s$ with $s = 576$, we get $0 = -16t^2 + 576$. Solving for $t$, we find $t = 6$ seconds.

Question 13

A rock falls from a tower that is 480 feet high. As it is falling, its height is given by the function $h(t)=-16 t^{2}+480 t$. How many seconds will it take for the rock to land on the ground?
A) 21.5 seconds
B) 14,400 seconds
C) 21.9 seconds
D) 5.5 seconds

Accepted Answer

To find the time it takes for the rock to hit the ground, we set $h(t) = 0$ and solve for $t$. This gives us the equation $0 = -16t^2 + 480$. Dividing through by -16 gives $t^2 = 30$, and taking the square root gives $t = \sqrt{30}$, which is approximately 5.5 seconds.

Quiz 5: Exponents, Polynomials, and Factoring Polynomials

A rug is to fit in a room so that a border of even width is left on all four sides. If the room is 17 feet by 22 feet and the area of the rug is 204 square feet, how wide will the border be?

A rectangular garden has dimensions of 18 feet by 15 feet. A gravel path of equal width is to be built around the garden. How wide can the path be if there is enough gravel for 234 square feet?

The area of a square lawn is numerically 12 feet more than the perimeter. Find the length of the side.

A rectangular space of 390 square feet is allocated for the living and dining areas in an apartment. Find the width of the square living area given that the width of the dining area is 11 feet.

The height of a triangle is 4 centimeters more than the base. If the area of the triangle is 126 square centimeters, find the base and height of the triangle.

If a ball is thrown upward from the ground with an initial velocity of 48 feet per second, its height after $t$ seconds is given by the function $h(t) =-16 t^{2}+48 t$ . Find the number of seconds before the ball lands on the ground.

If a projectile is launched upward from the ground with an initial velocity of 64.5 feet per second, its height $h$ in feet $t$ seconds later is given by the function $h(t) =-16 t^{2}+64.5 t$ . After how many seconds does the projectile land on the ground?

If a projectile is launched upward from a height of 32 feet at an initial velocity of 80 feet per second, then its height after $t$ seconds is given by the function $h(t) =-16 t^{2}+80 t+32$ , where $h$ is in feet. After how many seconds will the projectile reach a height of 132 feet?

If a projectile is launched upward from a height of 96 feet at an initial velocity of 80 feet per second, then its height $h$ after $t$ seconds is given by the function $h(t) =-16 t^{2}+80 t+96$ . After how many seconds does the projectile land on the ground?

A projectile is launched upward from an initial height of 400 feet, and it lands on the ground after 8 seconds. What was the initial velocity of the projectile?

The height, in feet, of a free-falling object t seconds after being dropped from an initial height s can be found using the function $h(t) =-16 t^{2}+s$ . An object is dropped from a helicopter 576 feet above the ground. How long will it take the object to land on the ground?

A rock falls from a tower that is 480 feet high. As it is falling, its height is given by the function $h(t) =-16 t^{2}+480 t$ . How many seconds will it take for the rock to land on the ground?

Quiz 5: Exponents, Polynomials, and Factoring Polynomials

A rug is to fit in a room so that a border of even width is left on all four sides. If the room is 17 feet by 22 feet and the area of the rug is 204 square feet, how wide will the border be?

A rectangular garden has dimensions of 18 feet by 15 feet. A gravel path of equal width is to be built around the garden. How wide can the path be if there is enough gravel for 234 square feet?

The area of a square lawn is numerically 12 feet more than the perimeter. Find the length of the side.

A rectangular space of 390 square feet is allocated for the living and dining areas in an apartment. Find the width of the square living area given that the width of the dining area is 11 feet.

The height of a triangle is 4 centimeters more than the base. If the area of the triangle is 126 square centimeters, find the base and height of the triangle.

A triangle is 6 centimeters wider than it is tall. The area is 140 square cm\mathrm{cm}cm . Find the width (length of the base) .

If a ball is thrown upward from the ground with an initial velocity of 48 feet per second, its height after ttt seconds is given by the function h(t) =−16t2+48th(t) =-16 t^{2}+48 th(t) =−16t2+48t . Find the number of seconds before the ball lands on the ground.

If a projectile is launched upward from the ground with an initial velocity of 64.5 feet per second, its height hhh in feet ttt seconds later is given by the function h(t) =−16t2+64.5th(t) =-16 t^{2}+64.5 th(t) =−16t2+64.5t . After how many seconds does the projectile land on the ground?

If a projectile is launched upward from a height of 96 feet at an initial velocity of 80 feet per second, then its height hhh after ttt seconds is given by the function h(t) =−16t2+80t+96h(t) =-16 t^{2}+80 t+96h(t) =−16t2+80t+96 . After how many seconds does the projectile land on the ground?

A projectile is launched upward from an initial height of 400 feet, and it lands on the ground after 8 seconds. What was the initial velocity of the projectile?

A rock falls from a tower that is 480 feet high. As it is falling, its height is given by the function h(t) =−16t2+480th(t) =-16 t^{2}+480 th(t) =−16t2+480t . How many seconds will it take for the rock to land on the ground?

If a ball is thrown upward from the ground with an initial velocity of 48 feet per second, its height after $t$ seconds is given by the function $h(t) =-16 t^{2}+48 t$ . Find the number of seconds before the ball lands on the ground.

If a projectile is launched upward from the ground with an initial velocity of 64.5 feet per second, its height $h$ in feet $t$ seconds later is given by the function $h(t) =-16 t^{2}+64.5 t$ . After how many seconds does the projectile land on the ground?

If a projectile is launched upward from a height of 96 feet at an initial velocity of 80 feet per second, then its height $h$ after $t$ seconds is given by the function $h(t) =-16 t^{2}+80 t+96$ . After how many seconds does the projectile land on the ground?

A rock falls from a tower that is 480 feet high. As it is falling, its height is given by the function $h(t) =-16 t^{2}+480 t$ . How many seconds will it take for the rock to land on the ground?