Question 1

A lot is in the shape of a right triangle. The shorter leg measures 90 m. The hypotenuse is 30 m longer than the length of the longer leg. How long is the longer leg?

Accepted Answer

Let the length of the longer leg be $x$ meters. According to the Pythagorean theorem, $90^2 + x^2 = (x + 30)^2$. Solving this equation gives $x = 120$ meters.

Question 2

A rock falls from a tower that is 208 feet high. As it is falling, its height is given by the formula $h = 208 - 16 t { } ^ { 2 }$. How many seconds (in tenths) will it take for the rock to hit the ground $( h = 0 )$ ?

Accepted Answer

To find the time it takes for the rock to hit the ground, set $h = 0$ and solve for $t$: $0 = 208 - 16t^2$. Solving for $t^2$ gives $t^2 = 13$, and taking the square root gives $t = \sqrt{13} \approx 3.6$ seconds.

Question 3

The height of a box is 5 inches. Its length is 5 inches more than its width. Find the length if the volume is 120.

Accepted Answer

The volume of a box is given by the formula length × width × height. Let the width be x inches. Then, the length is x + 5 inches. Given the height is 5 inches and the volume is 120 cubic inches, we have the equation: 5(x)(x + 5) = 120. Simplifying gives x^2 + 5x - 24 = 0, which factors to (x + 8)(x - 3) = 0. Thus, x = 3 inches (width). The length is x + 5 = 3 + 5 = 8 inches.

Question 4

A square lawn has an area of 1152 square feet. A sprinkler placed at the center of the lawn sprays a water in a circular pattern that just covers the lawn. What is the radius of the circle?

Accepted Answer

The answer of A square lawn has an area of...

Question 5

A boat is 180 feet from the base of cliff. If the distance from the top of the cliff to the boat is 45 more than twice the height of the cliff. Find the height of the cliff. Round to the nearest tenth of a foot if necessary.

Accepted Answer

Using the Pythagorean theorem, let h be the height of the cliff. The distance from the top of the cliff to the boat is 2h + 45. So, (2h + 45)^2 = h^2 + 180^2. Solving for h gives h = 75 ft.

Question 6

The area of a square is numerically 4 less than the perimeter. Find the length of the side, if the side is greater than 1.

Accepted Answer

The answer of The area of a square is numerically...

Question 7

A can has a surface area of 790 square inches. Its height is 6.91 inches. What is the radius of the circular top? Round to the nearest hundredth.

Accepted Answer

The surface area of a cylinder is given by the formula $2\pi r^2 + 2\pi rh$, where $r$ is the radius and $h$ is the height. Given the surface area (790 square inches) and the height (6.91 inches), we can solve for $r$. Rearranging the formula to solve for $r$ and substituting the given values, we find that the radius is approximately 8.28 inches when rounded to the nearest hundredth.

Question 8

A rock falls from a tower that is $117.6 \mathrm {~m}$ high. As it is falling, its height is given by the formula $h = 117.6 = 4.9 \mathrm { t } 2$, with $t$ in seconds. How many seconds will it take for the rock to hit the ground $( h = 0 )$ ?

Accepted Answer

The correct formula should be $h = 117.6 - 4.9t^2$. Setting $h = 0$, we solve $117.6 - 4.9t^2 = 0$, which gives $t^2 = 24$, so $t = \sqrt{24} \approx 4.9$ seconds.

Question 9

To solve for the lengths of the right triangle sides, which equation is correct?

Accepted Answer

The answer of To solve for the lengths of the...

Question 10

The position of an object moving in a straight line is given by s $s = 2 t ^ { 2 } - 3 t$ 3t, where s is in meters and t is the time in seconds the object has been in motion. How long (to the nearest tenth) will it take the object to move 9
Meters?

Accepted Answer

The answer of The position of an object moving in...

Question 11

A building needs a ramp to make it handicap accessible. By law, the ramp must run 10 inches horizontally for every 1 inch of rise. If the surface of the ramp is 52 inches long how far above the ground is the inclined end of
The ramp? Round to the nearest tenth of an inch.

Accepted Answer

The answer of A building needs a ramp to make...

Question 12

A ball is dropped from a cliff that is 368 feet high. The distance S (in feet) that it falls in t seconds is given by the formula S = 16t2. How many seconds (to tenths) will it take for the ball to hit the ground?

Accepted Answer

To find the time it takes for the ball to hit the ground, we use the given formula S = 16t^2 and set S = 368 feet. Solving for t, we get t = sqrt(368/16) = sqrt(23) ≈ 4.8 seconds.

Question 13

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 12 feet by 15 feet and the area of the rug is 108 square feet, how wide will the border be?

Accepted Answer

The area of the room is 12 ft * 15 ft = 180 square feet. The area of the rug is 108 square feet, so the area of the border is 180 - 108 = 72 square feet. Let x be the width of the border. The dimensions of the rug will then be (12 - 2x) by (15 - 2x), and its area is (12 - 2x)(15 - 2x) = 108. Solving for x, we find that x = 1.5 ft.

Question 14

If a rectangle is x feet long and y feet wide, which one of the following expressions is the length of its diagonal in terms of x and y?

Accepted Answer

The length of the diagonal of a rectangle can be found using the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Here, the diagonal acts as the hypotenuse, and the sides of the rectangle are the other two sides. Therefore, the length of the diagonal is $\sqrt{x^2 + y^2}$.

Question 15

A boat is 84 feet from the base of cliff. If the distance from the top of the cliff to the boat is 21 less than twice the height of the cliff to the water. Find the height of the cliff. Round to the nearest tenth of a foot if necessary.

Accepted Answer

The distance from the top of the cliff to the boat is 21 less than twice the height of the cliff to the water. So, the height of the cliff is (84 + 21) / 2 = 63 feet.

Question 16

The height of a box is 7 inches. The length is three inches more than the width. Find the width if the volume is 126.

Accepted Answer

The answer of The height of a box is 7...

Question 17

A ball is thrown downward from a window in a tall building. Its position at time t in seconds is $s = 16 t ^ { 2 } + 32 t$ where s is in feet. How long (to the nearest tenth) will it take the ball to fall 133 feet?

Accepted Answer

To find the time it takes for the ball to fall 133 feet, set $s = 133$ and solve the equation $16t^2 + 32t = 133$. Solving this quadratic equation gives $t \approx 2.1$ seconds.

Question 18

A 16 ft by 18 ft rectangular garden is to have a gravel path of uniform width bordering it. How wide is the path if the total area covered by the garden and path is 440 $\mathrm { ft } ^ { 2 }$ ?

Accepted Answer

The width of the path can be found by setting up an equation based on the area. The area of the garden is 16 ft * 18 ft = 288 ft^2. If the total area including the path is 440 ft^2, then the area of the path alone is 440 ft^2 - 288 ft^2 = 152 ft^2. Let x be the width of the path. The overall dimensions of the garden plus the path are (16 + 2x) by (18 + 2x). The equation for the total area is then (16 + 2x)(18 + 2x) = 440. Simplifying this equation will eventually lead to finding that x = 2, meaning the path is 2 ft wide.

Question 19

A ladder is resting against a wall. The top of the ladder touches the wall at a height of 12 ft. Find the length of the ladder if the length is 4 ft more than its distance from the wall.

Accepted Answer

The problem can be solved using the Pythagorean theorem. Let the distance from the wall be x feet. Then, the length of the ladder is x + 4 feet. The equation is x^2 + 12^2 = (x + 4)^2. Solving for x gives x = 16, so the length of the ladder is 20 feet.

Question 20

Two cars leave an intersection. One car travels north; the other east. When the car traveling north had gone 12 mi, the distance between the cars was 4 mi more than the distance traveled by the car heading east. How far had
The east bound car traveled?

Accepted Answer

Using the Pythagorean theorem, the distance between the cars is the hypotenuse of a right triangle with legs of 12 miles and the distance traveled by the eastbound car (x). The equation is: $ \sqrt{12^2 + x^2} = x + 4 $. Solving this equation gives x = 16 miles.

Quiz 2: Equations and Inequalities

A lot is in the shape of a right triangle. The shorter leg measures 90 m. The hypotenuse is 30 m longer than the length of the longer leg. How long is the longer leg?

A rock falls from a tower that is 208 feet high. As it is falling, its height is given by the formula $h = 208 - 16 t { } ^ { 2 }$ . How many seconds (in tenths) will it take for the rock to hit the ground $( h = 0 )$ ?

The height of a box is 5 inches. Its length is 5 inches more than its width. Find the length if the volume is 120.

A square lawn has an area of 1152 square feet. A sprinkler placed at the center of the lawn sprays a water in a circular pattern that just covers the lawn. What is the radius of the circle?

A boat is 180 feet from the base of cliff. If the distance from the top of the cliff to the boat is 45 more than twice the height of the cliff. Find the height of the cliff. Round to the nearest tenth of a foot if necessary.

The area of a square is numerically 4 less than the perimeter. Find the length of the side, if the side is greater than 1.

A can has a surface area of 790 square inches. Its height is 6.91 inches. What is the radius of the circular top? Round to the nearest hundredth.

A rock falls from a tower that is $117.6 \mathrm {~m}$ high. As it is falling, its height is given by the formula $h = 117.6 = 4.9 \mathrm { t } 2$ , with $t$ in seconds. How many seconds will it take for the rock to hit the ground $( h = 0 )$ ?

The position of an object moving in a straight line is given by s $s = 2 t ^ { 2 } - 3 t$ 3t, where s is in meters and t is the time in seconds the object has been in motion. How long (to the nearest tenth) will it take the object to move 9 Meters?

A building needs a ramp to make it handicap accessible. By law, the ramp must run 10 inches horizontally for every 1 inch of rise. If the surface of the ramp is 52 inches long how far above the ground is the inclined end of The ramp? Round to the nearest tenth of an inch.

A ball is dropped from a cliff that is 368 feet high. The distance S (in feet) that it falls in t seconds is given by the formula S = 16t2. How many seconds (to tenths) will it take for the ball to hit the ground?

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 12 feet by 15 feet and the area of the rug is 108 square feet, how wide will the border be?

If a rectangle is x feet long and y feet wide, which one of the following expressions is the length of its diagonal in terms of x and y?

A boat is 84 feet from the base of cliff. If the distance from the top of the cliff to the boat is 21 less than twice the height of the cliff to the water. Find the height of the cliff. Round to the nearest tenth of a foot if necessary.

The height of a box is 7 inches. The length is three inches more than the width. Find the width if the volume is 126.

A ball is thrown downward from a window in a tall building. Its position at time t in seconds is $s = 16 t ^ { 2 } + 32 t$ where s is in feet. How long (to the nearest tenth) will it take the ball to fall 133 feet?

A 16 ft by 18 ft rectangular garden is to have a gravel path of uniform width bordering it. How wide is the path if the total area covered by the garden and path is 440 $\mathrm { ft } ^ { 2 }$ ?

A ladder is resting against a wall. The top of the ladder touches the wall at a height of 12 ft. Find the length of the ladder if the length is 4 ft more than its distance from the wall.

Two cars leave an intersection. One car travels north; the other east. When the car traveling north had gone 12 mi, the distance between the cars was 4 mi more than the distance traveled by the car heading east. How far had The east bound car traveled?

Quiz 2: Equations and Inequalities

A lot is in the shape of a right triangle. The shorter leg measures 90 m. The hypotenuse is 30 m longer than the length of the longer leg. How long is the longer leg?

A rock falls from a tower that is 208 feet high. As it is falling, its height is given by the formula h=208−16t2h = 208 - 16 t { } ^ { 2 }h=208−16t2 . How many seconds (in tenths) will it take for the rock to hit the ground (h=0) ( h = 0 ) (h=0) ?

The height of a box is 5 inches. Its length is 5 inches more than its width. Find the length if the volume is 120.

A square lawn has an area of 1152 square feet. A sprinkler placed at the center of the lawn sprays a water in a circular pattern that just covers the lawn. What is the radius of the circle?

A boat is 180 feet from the base of cliff. If the distance from the top of the cliff to the boat is 45 more than twice the height of the cliff. Find the height of the cliff. Round to the nearest tenth of a foot if necessary.

The area of a square is numerically 4 less than the perimeter. Find the length of the side, if the side is greater than 1.

A can has a surface area of 790 square inches. Its height is 6.91 inches. What is the radius of the circular top? Round to the nearest hundredth.

A rock falls from a tower that is 117.6 m117.6 \mathrm {~m}117.6 m high. As it is falling, its height is given by the formula h=117.6=4.9t2h = 117.6 = 4.9 \mathrm { t } 2h=117.6=4.9t2 , with ttt in seconds. How many seconds will it take for the rock to hit the ground (h=0) ( h = 0 ) (h=0) ?

To solve for the lengths of the right triangle sides, which equation is correct?

The position of an object moving in a straight line is given by s s=2t2−3ts = 2 t ^ { 2 } - 3 ts=2t2−3t 3t, where s is in meters and t is the time in seconds the object has been in motion. How long (to the nearest tenth) will it take the object to move 9 Meters?

A building needs a ramp to make it handicap accessible. By law, the ramp must run 10 inches horizontally for every 1 inch of rise. If the surface of the ramp is 52 inches long how far above the ground is the inclined end of The ramp? Round to the nearest tenth of an inch.

A ball is dropped from a cliff that is 368 feet high. The distance S (in feet) that it falls in t seconds is given by the formula S = 16t2. How many seconds (to tenths) will it take for the ball to hit the ground?

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 12 feet by 15 feet and the area of the rug is 108 square feet, how wide will the border be?

If a rectangle is x feet long and y feet wide, which one of the following expressions is the length of its diagonal in terms of x and y?

A boat is 84 feet from the base of cliff. If the distance from the top of the cliff to the boat is 21 less than twice the height of the cliff to the water. Find the height of the cliff. Round to the nearest tenth of a foot if necessary.

The height of a box is 7 inches. The length is three inches more than the width. Find the width if the volume is 126.

A ball is thrown downward from a window in a tall building. Its position at time t in seconds is s=16t2+32ts = 16 t ^ { 2 } + 32 ts=16t2+32t where s is in feet. How long (to the nearest tenth) will it take the ball to fall 133 feet?

A 16 ft by 18 ft rectangular garden is to have a gravel path of uniform width bordering it. How wide is the path if the total area covered by the garden and path is 440 ft2\mathrm { ft } ^ { 2 }ft2 ?

A ladder is resting against a wall. The top of the ladder touches the wall at a height of 12 ft. Find the length of the ladder if the length is 4 ft more than its distance from the wall.

Two cars leave an intersection. One car travels north; the other east. When the car traveling north had gone 12 mi, the distance between the cars was 4 mi more than the distance traveled by the car heading east. How far had The east bound car traveled?

A rock falls from a tower that is 208 feet high. As it is falling, its height is given by the formula $h = 208 - 16 t { } ^ { 2 }$ . How many seconds (in tenths) will it take for the rock to hit the ground $( h = 0 )$ ?

A rock falls from a tower that is $117.6 \mathrm {~m}$ high. As it is falling, its height is given by the formula $h = 117.6 = 4.9 \mathrm { t } 2$ , with $t$ in seconds. How many seconds will it take for the rock to hit the ground $( h = 0 )$ ?

The position of an object moving in a straight line is given by s $s = 2 t ^ { 2 } - 3 t$ 3t, where s is in meters and t is the time in seconds the object has been in motion. How long (to the nearest tenth) will it take the object to move 9 Meters?

A ball is thrown downward from a window in a tall building. Its position at time t in seconds is $s = 16 t ^ { 2 } + 32 t$ where s is in feet. How long (to the nearest tenth) will it take the ball to fall 133 feet?

A 16 ft by 18 ft rectangular garden is to have a gravel path of uniform width bordering it. How wide is the path if the total area covered by the garden and path is 440 $\mathrm { ft } ^ { 2 }$ ?