Question 1

Solve the problem.
-A ball is dropped from a cliff that is 256 feet high. The distance $S$ (in feet) that it falls in $t$ seconds is given by the formula $S = 16 t ^ { 2 }$. 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 set $S = 256$ feet and solve for $t$ in the equation $S = 16t^2$. Substituting $S = 256$, we get $256 = 16t^2$. Dividing both sides by 16, we get $t^2 = 16$. Taking the square root of both sides gives $t = 4$ seconds.

Question 2

Solve the problem.
-A rock falls from a tower that is $68.6 \mathrm {~m}$ high. As it is falling, its height is given by the formula $h = 68.6 - 4.9 t ^ { 2 }$, with $t$ in seconds. How many seconds 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 $t$: $0 = 68.6 - 4.9t^2$. Solving this equation gives $t = 3.7$ seconds.

Question 3

Solve the problem.
-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 73 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 ramp runs 10 inches horizontally for every 1 inch of rise. So, if the surface of the ramp is 73 inches long, the inclined end of the ramp is 73/10 = 7.3 inches above the ground.

Question 4

Solve the problem.
-Your company uses the quadratic model y = -11x2 + 350x to represent how many units (y) of a new product will be sold (x) weeks after its release . How many units can you expect to sell in
Week 20?

Accepted Answer

To find the number of units sold in Week 20, substitute x = 20 into the quadratic model: y = -11(20)^2 + 350(20). This simplifies to y = -11(400) + 7000 = -4400 + 7000 = 2600 units.

Question 5

Solve the problem.
-The stadium vending company finds that sales of hot dogs average 50,000 hot dogs per game when the hot dogs sell for $\$ 2.50$ each. For each 50 cent increase in the price, the sales per game drop by 5000 hot dogs. What price per hot dog should the vending company charge to realize the maximum revenue?

Accepted Answer

To maximize revenue, we need to find the price per hot dog that results in the highest product of price and quantity sold. Let $x$ represent the number of 50 cent increases in the price. The price per hot dog can be represented as $2.50 + 0.50x$ dollars, and the quantity sold can be represented as $50000 - 5000x$ hot dogs. The revenue function, $R$, is the product of the price and quantity, so $R(x) = (2.50 + 0.50x)(50000 - 5000x)$. Simplifying, we get $R(x) = 125000 - 12500x + 25000x - 2500x^2$, or $R(x) = -2500x^2 + 12500x + 125000$. To find the maximum revenue, we need the vertex of this parabola, which occurs at $x = -b/2a$. Here, $a = -2500$ and $b = 12500$, so $x = -12500/(2*(-2500)) = 2.5$. This means the price increase should be $2.5 * 0.50 = \$1.25$, making the optimal price $2.50 + 1.25 = \$3.75$.

Question 6

Solve the problem.
-A ladder is resting against a wall. The top of the ladder touches the wall at a height of 6 ft. Find the length of the ladder if the length is 2 ft more than its distance from the wall.

Accepted Answer

Using the Pythagorean theorem, let the distance from the wall be x. Then, the ladder's length is x + 2. So, x^2 + 6^2 = (x + 2)^2. Solving gives x = 8, so the ladder is 10 ft long.

Question 7

Solve the problem.
-A rock falls from a tower that is 176 feet high. As it is falling, its height is given by the formula $h = 176 - 16 t ^ { 2 }$. 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 $t$: $0 = 176 - 16t^2$. Rearranging gives $16t^2 = 176$, and dividing both sides by 16 gives $t^2 = 11$. Taking the square root of both sides gives $t = \sqrt{11}$, which is approximately $3.3$ seconds.

Question 8

Solve the problem.
-A can has a surface area of 939 square inches. Its height is 6.64 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 (939 square inches) and the height (6.64 inches), we can solve for the radius. Rearranging the formula to solve for $r$ and substituting the given values, we find that the radius is approximately 9.35 inches when rounded to the nearest hundredth.

Question 9

Solve the problem.
-A lot is in the shape of a right triangle. The shorter leg measures 120 m. The hypotenuse is 40 m longer than the length of the longer leg. How long is the longer leg?

Accepted Answer

Let the longer leg be x meters. According to the problem, the hypotenuse is x + 40 meters. Using the Pythagorean theorem: $120^2 + x^2 = (x + 40)^2$. Solving this equation, we find that x = 160 m.

Question 10

Solve the problem.
-Two cars leave an intersection. One car travels north; the other east. When the car traveling north had gone 18 mi, the distance between the cars was 6 mi more than the distance traveled by the car
Heading east. How far had the east bound car traveled?

Accepted Answer

The problem can be solved using the Pythagorean theorem, where the distance between the two cars forms the hypotenuse of a right triangle. Let $x$ be the distance traveled by the eastbound car. According to the problem, the hypotenuse (distance between the cars) is $x + 6$ miles. The northbound car has traveled 18 miles, so we have $18^2 + x^2 = (x + 6)^2$. Solving this equation for $x$ gives $x = 24$ miles.

Question 11

Solve the problem.
-A boat is 108 feet from the base of cliff. If the distance from the top of the cliff to the boat is 27 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 answer of Solve the problem.
-A boat is 108 feet...

Question 12

Solve the problem.
-A square lawn has an area of 2450 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 area of the square lawn is 2450 square feet, so each side of the square is √2450 feet. The diagonal of the square, which is the diameter of the circle, is √2 times the side length. Therefore, the radius of the circle is (√2 * √2450) / 2 = 35 ft.

Question 13

Answer the question.
-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. Therefore, if a rectangle has sides of lengths $x$ and $y$, the length of its diagonal $d$ is given by $d = \sqrt{x^2 + y^2}$.

Question 14

Solve the problem.
-The position of an object moving in a straight line is given by $s = 2 t ^ { 2 } - 3 t$, 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 5 meters?

Accepted Answer

The answer of Solve the problem.
-The position of an object...

Question 15

Solve the problem.
-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 152 feet?

Accepted Answer

To find the time it takes for the ball to fall 152 feet, set $s = 152$ and solve for $t$: $152 = 16t^2 + 32t$. Rearranging gives $16t^2 + 32t - 152 = 0$. Dividing the entire equation by 16 gives $t^2 + 2t - 9.5 = 0$. Using the quadratic formula, $t = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-9.5)}}{2(1)}$, which simplifies to $t = \frac{-2 \pm \sqrt{4 + 38}}{2}$, or $t = \frac{-2 \pm \sqrt{42}}{2}$. The positive solution is approximately $2.2$ seconds.

Question 16

Solve the problem.
-A boat is 144 feet from the base of cliff. If the distance from the top of the cliff to the boat is 36 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 36 + 2h. So, (36 + 2h)^2 = h^2 + 144^2. Solving for h gives h = 60 ft.

Question 17

Solve the problem.
-Your company uses the quadratic model $y = - 4.5 x ^ { 2 } + 150 x$ to represent the average number of new customers who will be signed on (x) weeks after the release of your new service. How many new customers can you expect to gain in week 22?

Accepted Answer

To find the number of new customers in week 22, substitute $x = 22$ into the quadratic model $y = - 4.5 x ^ { 2 } + 150 x$. This gives $y = - 4.5(22)^2 + 150(22) = - 4.5(484) + 3300 = - 2178 + 3300 = 1122$ customers.

Question 18

Answer the question.
-To solve for the lengths of the right triangle sides, which equation is correct?

Accepted Answer

The answer of Answer the question.
-To solve for the lengths...

Question 19

Solve the problem.
-If an amount of money, called the principal, $\mathrm { P }$, is deposited into an account that earns interest at a rate $r$, compounded annually, then in two years that investment will grow to an amount $A$, given by the formula $A = P ( 1 + r ) ^ { 2 }$. If a principal amount of $\$ 5500$ grows to $\$ 7147.80$ in two years, what is the interest rate?

Accepted Answer

To find the interest rate, we use the formula $A = P(1 + r)^2$ with $A = 7147.80$ and $P = 5500$. Rearranging the formula to solve for $r$, we get $r = \sqrt{\frac{A}{P}} - 1$. Substituting the given values, we find $r = \sqrt{\frac{7147.80}{5500}} - 1$, which simplifies to $r = \sqrt{1.2996} - 1$. This gives $r \approx 0.14$ or $14\%$.

Question 20

Decide what values of the variable cannot possibly be solutions for the equation.
-$$\frac { 1 } { 2 x } + \frac { 6 } { x } = 11$$

Accepted Answer

The variable $x$ cannot be 0 because it would make the denominators in the equation undefined, as division by zero is not possible.

Quiz 1: Equations and Inequalities

Solve the problem. -A ball is dropped from a cliff that is 256 feet high. The distance $S$ (in feet) that it falls in $t$ seconds is given by the formula $S = 16 t ^ { 2 }$ . How many seconds (to tenths) will it take for the ball to hit the ground?

Solve the problem. -A rock falls from a tower that is $68.6 \mathrm {~m}$ high. As it is falling, its height is given by the formula $h = 68.6 - 4.9 t ^ { 2 }$ , with $t$ in seconds. How many seconds will it take for the rock to hit the ground (h $= 0$ ) ?

Solve the problem. -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 73 inches long how far above the Ground is the inclined end of the ramp? Round to the nearest tenth of an inch.

Solve the problem. -Your company uses the quadratic model y = -11x2 + 350x to represent how many units (y) of a new product will be sold (x) weeks after its release . How many units can you expect to sell in Week 20?

Solve the problem. -A ladder is resting against a wall. The top of the ladder touches the wall at a height of 6 ft. Find the length of the ladder if the length is 2 ft more than its distance from the wall.

Solve the problem. -A rock falls from a tower that is 176 feet high. As it is falling, its height is given by the formula $h = 176 - 16 t ^ { 2 }$ . How many seconds (in tenths) will it take for the rock to hit the ground $( h = 0 )$ ?

Solve the problem. -A can has a surface area of 939 square inches. Its height is 6.64 inches. What is the radius of the circular top? Round to the nearest hundredth.

Solve the problem. -A lot is in the shape of a right triangle. The shorter leg measures 120 m. The hypotenuse is 40 m longer than the length of the longer leg. How long is the longer leg?

Solve the problem. -Two cars leave an intersection. One car travels north; the other east. When the car traveling north had gone 18 mi, the distance between the cars was 6 mi more than the distance traveled by the car Heading east. How far had the east bound car traveled?

Solve the problem. -A boat is 108 feet from the base of cliff. If the distance from the top of the cliff to the boat is 27 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.

Solve the problem. -A square lawn has an area of 2450 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?

Answer the question. -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 $y$ ?

Solve the problem. -The position of an object moving in a straight line is given by $s = 2 t ^ { 2 } - 3 t$ , 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 5 meters?

Solve the problem. -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 152 feet?

Solve the problem. -A boat is 144 feet from the base of cliff. If the distance from the top of the cliff to the boat is 36 more than twice the height of the cliff. Find the height of the cliff. Round to the nearest tenth of a Foot if necessary.

Solve the problem. -Your company uses the quadratic model $y = - 4.5 x ^ { 2 } + 150 x$ to represent the average number of new customers who will be signed on (x) weeks after the release of your new service. How many new customers can you expect to gain in week 22?

Decide what values of the variable cannot possibly be solutions for the equation. - $\frac { 1 } { 2 x } + \frac { 6 } { x } = 11$

Quiz 1: Equations and Inequalities

Solve the problem. -A ball is dropped from a cliff that is 256 feet high. The distance SSS (in feet) that it falls in ttt seconds is given by the formula S=16t2S = 16 t ^ { 2 }S=16t2 . How many seconds (to tenths) will it take for the ball to hit the ground?

Solve the problem. -A rock falls from a tower that is 68.6 m68.6 \mathrm {~m}68.6 m high. As it is falling, its height is given by the formula h=68.6−4.9t2h = 68.6 - 4.9 t ^ { 2 }h=68.6−4.9t2 , with ttt in seconds. How many seconds will it take for the rock to hit the ground (h =0= 0=0 ) ?