Question 1

Solve the problem
-Bob owns a watch repair shop. He has found that the cost of operating his shop is given by $c ( x ) = 4 x ^ { 2 } - 208 x + 49$, where $c$ is cost and $x$ is the number of watches repaired. How many watches must he repair to have the lowest cost?

Accepted Answer

The lowest cost occurs at the vertex of the parabola represented by the cost function $c(x) = 4x^2 - 208x + 49$. The x-coordinate of the vertex can be found using the formula $-\frac{b}{2a}$, where $a = 4$ and $b = -208$. Calculating gives $-\frac{-208}{2*4} = 26$, meaning Bob must repair 26 watches to have the lowest cost.

Question 2

Solve the problem
-John owns a hot dog stand. He has found that his profit is represented by the equation $\mathrm { P } ( \mathrm { x } ) = - \mathrm { x } ^ { 2 } + 70 \mathrm { x } + 82$, with $\mathrm { P }$ being profits and $\mathrm { x }$ the number of hot dogs sold. How many hot dogs must he sell to earn the most profit?

Accepted Answer

The equation for profit, $P(x) = -x^2 + 70x + 82$, is a quadratic equation in the form of $ax^2 + bx + c$, where the coefficient of $x^2$ is negative, indicating a downward-opening parabola. The vertex of this parabola, given by the formula $x = -\frac{b}{2a}$, represents the point of maximum profit. Here, $a = -1$ and $b = 70$, so $x = -\frac{70}{2(-1)} = 35$. Therefore, John must sell 35 hot dogs to earn the most profit.

Question 3

Solve the problem
-If an object is propelled upward from a height of 112 feet at an initial velocity of 96 feet per second, then its height after $t$ seconds is given by the equation $h ( t ) = - 16 t ^ { 2 } + 96 t + 112$. After how many seconds does the object hit the ground?

Accepted Answer

The object hits the ground when $h(t) = 0$. Solving $-16t^2 + 96t + 112 = 0$ using the quadratic formula gives $t = 7$ seconds.

Question 4

Solve the problem
-A rock is propelled upward from the top of a building 180 feet tall at an initial velocity of 56 feet per second. The function that describes the height of the rocket in terms of time $t$ is $\mathrm { s } ( \mathrm { t } ) = - 16 \mathrm { t } ^ { 2 } + 56 t + 180$. Determine the maximum height that the rock reaches.

Accepted Answer

The answer of Solve the problem
-A rock is propelled upward...

Question 5

Solve the problem
-A rock is propelled upward from the top of a building 160 feet tall at an initial velocity of 200 feet per second. Give the function that describes the height of the rocket in terms of time $t$.

Accepted Answer

The correct function that describes the height of the rocket in terms of time $t$ is given by the formula $f(t) = -16t^2 + vt + h$, where $v$ is the initial velocity and $h$ is the initial height. Here, $v = 200$ feet per second and $h = 160$ feet, so the correct function is $f(t) = -16t^2 + 200t + 160$.

Question 6

Solve the problem
-If an object is thrown upward with an initial velocity of $48 \mathrm { ft } / \mathrm { sec }$, its height after $\mathrm { t }$ sec is given by $\mathrm { s } ( \mathrm { t } ) = 48 \mathrm { t } - 16 \mathrm { t } ^ { 2 }$. Find the maximum height attained by the object. (The object will attain maximum height exactly at the halfway point in terms of the time $t$, where $t = 0$ is at the beginning of the object's flight, and the final time is when the object hits the ground.)

Accepted Answer

The answer of Solve the problem
-If an object is thrown...

Question 7

Solve the problem
-A farmer has 400 feet of fence with which to fence a rectangular plot of land. The plot lies along a river so that only three sides need to be fenced. Estimate the largest area that can be fenced.

Accepted Answer

The largest area that can be fenced is obtained when the plot is a rectangle with the length twice the width. Let the width be $w$ and the length be $2w$. The total length of the fence used for the two widths and one length is $400 = 2w + 2w$, so $400 = 4w$, giving $w = 100$ feet. The length is $2w = 200$ feet. Therefore, the area is $w \times 2w = 100 \times 200 = 20,000 \mathrm { ft } ^ { 2 }$.

Question 8

Solve the problem
-If a rocket is propelled upward from ground level, its height in meters after $\mathrm { t }$ seconds is given by $\mathrm { h } ( \mathrm { t } ) = - 9.8 \mathrm { t } ^ { 2 } + 107.8 \mathrm { t }$. During what interval of time will the rocket be higher than $294 \mathrm {~m} ?$

Accepted Answer

The answer of Solve the problem
-If a rocket is propelled...

Question 9

Solve the problem
-The number of mosquitoes $\mathrm { M } ( \mathrm { x } )$, in millions, in a certain area depends on the June rainfall $\mathrm { x }$, in inches, according to the equation $\mathrm { M } ( \mathrm { x } ) = 9 \mathrm { x } - \mathrm { x } ^ { 2 }$. What rainfall produces the maximum number of mosquitoes?

Accepted Answer

The maximum value of a quadratic function $f(x) = ax^2 + bx + c$ occurs at $x = -\frac{b}{2a}$. For $M(x) = -x^2 + 9x$, $a = -1$ and $b = 9$, so the maximum occurs at $x = -\frac{9}{2(-1)} = 4.5$.

Question 10

Solve the problem
-A coin is tossed upward from a balcony 176 feet high with an initial velocity of 16 feet per second, then its height after $t$ seconds is given by the equation $h ( t ) = - 16 t ^ { 2 } + 16 t + 176$. During what interval of time will the coin be at a height of at least $80 \mathrm { ft }$ ?

Accepted Answer

To find the interval of time when the coin is at least 80 ft high, we set the height equation equal to 80 and solve for $t$: $80 = -16t^2 + 16t + 176$. Rearranging gives $16t^2 - 16t - 96 = 0$, which simplifies to $t^2 - t - 6 = 0$. Factoring, we get $(t-3)(t+2) = 0$, so $t = 3$ or $t = -2$. Since time cannot be negative, we discard $t = -2$. The coin starts above 80 ft and reaches 80 ft again at $t = 3$ seconds, so the interval is $0 \leq t \leq 3$.

Question 11

Solve the problem
-The number of mosquitoes $\mathrm { M } ( \mathrm { x } )$, in millions, in a certain area depends on the June rainfall $\mathrm { x }$, in inches: $\mathrm { M } ( \mathrm { x } ) = 2 \mathrm { x } - \mathrm { x } ^ { 2 }$. What rainfall produces the maximum number of mosquitoes?

Accepted Answer

The maximum value of $M(x) = 2x - x^2$ can be found by completing the square or using the vertex formula for a parabola. The vertex of a parabola given by $ax^2 + bx + c$ is at $(-b/2a, M(-b/2a))$. Here, $a = -1$ and $b = 2$, so the x-coordinate of the vertex is $-2/(2*(-1)) = 1$. This means the maximum number of mosquitoes occurs at 1 inch of rainfall.

Question 12

Solve the problem
-Your company uses the quadratic model $y = - 11 x ^ { 2 } + 350 x$ to represent how many units (y) of a new product will be sold ( $\mathrm { x } )$ 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 13

Solve the problem
-A rock is propelled upward from the top of a building 210 feet tall at an initial velocity of 176 feet per second. The function that describes the height of the rocket in terms of time $t$ is $s ( t ) = - 16 t ^ { 2 } + 176 t + 210$. Determine the time at which the rock reaches its maximum height.

Accepted Answer

The maximum height is reached at the vertex of the parabola described by the function. The time at which this occurs is given by $ t = -\frac{b}{2a} $. Here, $ a = -16 $ and $ b = 176 $, so $ t = -\frac{176}{2(-16)} = 5.5 $ seconds.

Question 14

Solve the problem
-John owns a hotdog stand. His profit is represented by the equation $\mathrm { P } ( \mathrm { x } ) = - \mathrm { x } ^ { 2 } + 12 \mathrm { x } + 45$, with $\mathrm { P }$ being profits and $x$ the number of hotdogs sold. What is the most he can earn?

Accepted Answer

The profit function is a parabola with a maximum at the vertex. The x-coordinate of the vertex is -b/2a = -12/(-2) = 6. So, the most John can earn is P(6) = -6^2 + 12(6) + 45 = $81.

Question 15

Solve the problem
-If an object is propelled upward from a height of 96 feet at an initial velocity of 64 feet per second, then its height after $t$ seconds is given by the equation $h ( t ) = - 16 t ^ { 2 } + 64 t + 96$, where height is in feet. After how many seconds will the object reach a height of 160 feet?

Accepted Answer

To find when the object reaches 160 feet, substitute $h(t) = 160$ into the equation and solve for $t$: $160 = -16t^2 + 64t + 96$. Rearranging gives $0 = -16t^2 + 64t - 64$, which simplifies to $t^2 - 4t + 4 = 0$. Factoring yields $(t - 2)^2 = 0$, so $t = 2$ seconds.

Question 16

Solve the problem
-Of all possible pairs of positive numbers having a sum of 82, which pair has the maximum product?

Accepted Answer

The maximum product of two numbers with a fixed sum occurs when the two numbers are as close to each other as possible. In this case, that's 41 and 41, because 41 + 41 = 82, and the product is maximized when the numbers are equal or nearly equal.

Question 17

Solve the problem
-A ball is thrown downward from a window in a tall building. Its position at time $t$ in seconds is $s ( t ) = 16 t ^ { 2 } + 32 t$, where $s ( t )$ is in feet. How long (to the nearest tenth) will it take the ball to fall 140 feet?

Accepted Answer

To find the time it takes for the ball to fall 140 feet, set $s(t) = 140$ and solve for $t$: $16t^2 + 32t = 140$. Simplifying gives $16t^2 + 32t - 140 = 0$. Solving this quadratic equation using the quadratic formula gives $t \approx 2.1$ seconds.

Question 18

Solve the problem
-A toy rocket is shot vertically upward from the ground. Its distance in feet from the ground in $t$ seconds is given by $\mathrm { s } ( \mathrm { t } ) = - 16 \left( \mathrm { t } ^ { 2 } \right) + 142 \mathrm { t }$. At what time(s) will the ball be $151 \mathrm { ft }$ from the ground? Round your answer to the nearest tenth of a second.

Accepted Answer

To find when the ball will be 151 ft from the ground, set $s(t) = 151$ and solve for $t$: $151 = -16t^2 + 142t$. Rearranging gives $16t^2 - 142t + 151 = 0$. Solving this quadratic equation yields $t \approx 1.2$ and $t \approx 7.6$ seconds.

Question 19

Solve the problem
-The length of a table is 17 inches more than its width. If the area of the table is 2280 square inches, what is its length?

Accepted Answer

The answer of Solve the problem
-The length of a table...

Question 20

Solve the problem
-$x$ and $y$ are two positive numbers and $y$ is three greater than $x$. The product of the numbers is 340 . Find the smaller number, $x$.

Accepted Answer

Let $ y = x + 3 $. The product of the numbers is $ x(x + 3) = 340 $. Solving the quadratic equation $ x^2 + 3x - 340 = 0 $ using the quadratic formula gives $ x = 17 $ or $ x = -20 $. Since $ x $ is positive, the smaller number is 17.

Quiz 4: Polynomial and Rational Functions

Solve the problem -Bob owns a watch repair shop. He has found that the cost of operating his shop is given by $c ( x ) = 4 x ^ { 2 } - 208 x + 49$ , where $c$ is cost and $x$ is the number of watches repaired. How many watches must he repair to have the lowest cost?

Solve the problem -If an object is propelled upward from a height of 112 feet at an initial velocity of 96 feet per second, then its height after $t$ seconds is given by the equation $h ( t ) = - 16 t ^ { 2 } + 96 t + 112$ . After how many seconds does the object hit the ground?

Solve the problem -A rock is propelled upward from the top of a building 160 feet tall at an initial velocity of 200 feet per second. Give the function that describes the height of the rocket in terms of time $t$ .

Solve the problem -A farmer has 400 feet of fence with which to fence a rectangular plot of land. The plot lies along a river so that only three sides need to be fenced. Estimate the largest area that can be fenced.

Solve the problem -Your company uses the quadratic model $y = - 11 x ^ { 2 } + 350 x$ to represent how many units (y) of a new product will be sold ( $\mathrm { x } )$ weeks after its release. How many units can you expect to sell in week $20 ?$

Solve the problem -John owns a hotdog stand. His profit is represented by the equation $\mathrm { P } ( \mathrm { x } ) = - \mathrm { x } ^ { 2 } + 12 \mathrm { x } + 45$ , with $\mathrm { P }$ being profits and $x$ the number of hotdogs sold. What is the most he can earn?

Solve the problem -Of all possible pairs of positive numbers having a sum of 82, which pair has the maximum product?

Solve the problem -A ball is thrown downward from a window in a tall building. Its position at time $t$ in seconds is $s ( t ) = 16 t ^ { 2 } + 32 t$ , where $s ( t )$ is in feet. How long (to the nearest tenth) will it take the ball to fall 140 feet?

Solve the problem -The length of a table is 17 inches more than its width. If the area of the table is 2280 square inches, what is its length?

Solve the problem - $x$ and $y$ are two positive numbers and $y$ is three greater than $x$ . The product of the numbers is 340 . Find the smaller number, $x$ .

Quiz 4: Polynomial and Rational Functions

Solve the problem -A rock is propelled upward from the top of a building 160 feet tall at an initial velocity of 200 feet per second. Give the function that describes the height of the rocket in terms of time ttt .