Question 1

Solve the problem.
-$x$ and $y$ are two positive numbers and $y$ is three greater than $x$. The product of the numbers is 270 . Find the smaller number, $x$.
A) 15
B) 18
C) 9
D) 8

Accepted Answer

The answer of Solve the problem.
-$x$ and $y$ are two...

Question 2

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: $\mathrm { M } ( \mathrm { x } ) = 13 \mathrm { x } - \mathrm { x } ^ { 2 }$. What rainfall produces the maximum number of mosquitoes?
A) 0 in.
B) $169 \mathrm { in }$.
C) $6.5 \mathrm { in }$.
D) 13 in.

Accepted Answer

The maximum value of a quadratic function $f(x) = ax^2 + bx + c$ occurs at $x = -\frac{b}{2a}$. For $M(x) = -x^2 + 13x$, $a = -1$ and $b = 13$, so the maximum occurs at $x = -\frac{13}{2(-1)} = 6.5$.

Question 3

Solve the problem.
-The length of a table is 11 inches more than its width. If the area of the table is 1692 square inches, what is its length?
A) $25 \mathrm { in }$.
B) 47 in.
C) $72 \mathrm { in. }$
D) $36 \mathrm { in }$.

Accepted Answer

The equation for the area of the rectangle is $l \times w = 1692$, where $l = w + 11$. Substituting $l$ in the area equation gives $(w + 11)w = 1692$. Solving this quadratic equation, we find $w = 36$ inches, so $l = 36 + 11 = 47$ inches.

Question 4

Solve the problem.
-If an object is thrown upward with an initial velocity of $16 \mathrm { ft } / \mathrm { sec }$, its height after $\mathrm { t }$ sec is given by $s ( t ) = 16 t - 16 t ^ { 2 }$. 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.)
A) $16 \mathrm { ft }$
B) $4 \mathrm { ft }$
C) $- 12 \mathrm { ft }$
D) $0 \mathrm { ft }$

Accepted Answer

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

Question 5

Solve the problem.
-If a rocket is propelled upward from ground level, its height in meters after $t$ seconds is given by $h ( t ) = - 9.8 t ^ { 2 } + 98 t$. During what interval of time will the rocket be higher than $235.2 \mathrm {~m}$ ?
A) $6 < t < 8$
B) $8 < t < 10$
C) $0 < t < 4$
D) $4 < t < 6$

Accepted Answer

The rocket will be higher than 235.2 m when $h(t) > 235.2$. Solving this inequality gives $4 < t < 6$.

Question 6

Solve the problem.
-If an object is propelled 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 equation $\mathrm { h } ( \mathrm { t } ) = - 16 \mathrm { t } ^ { 2 } + 80 \mathrm { t } + 32$, where height is in feet. After how many seconds will the object reach a height of 132 feet?
A) $1.3 \mathrm { sec }$
B) $10 \mathrm { sec }$
C) $2.5 \mathrm { sec }$
D) $5 \mathrm { sec }$

Accepted Answer

To find when the object reaches a height of 132 feet, set $h(t) = 132$ and solve for $t$: $132 = -16t^2 + 80t + 32$. Rearranging gives $16t^2 - 80t + 100 = 0$, which simplifies to $t^2 - 5t + 6.25 = 0$. Factoring or using the quadratic formula yields $t = 2.5$ seconds.

Question 7

Solve the problem.
-The number of mosquitoes $M ( x )$, in millions, in a certain area depends on the June rainfall $x$, in inches, according to the equation $M ( x ) = 6 x - x ^ { 2 }$. What rainfall produces the maximum number of mosquitoes?
A) 3 in.
B) $6 \mathrm { in. }$
C) $0 \mathrm { in }$.
D) $36 \mathrm { in. }$

Accepted Answer

The maximum value of $M(x)$ occurs at the vertex of the parabola represented by the equation $M(x) = 6x - x^2$. The x-coordinate of the vertex can be found using the formula $-\frac{b}{2a}$ for a quadratic equation in the form $ax^2 + bx + c$. Here, $a = -1$ and $b = 6$, so the x-coordinate is $-\frac{6}{2(-1)} = 3$. This means the maximum number of mosquitoes occurs with 3 inches of rainfall.

Question 8

Solve the problem.
-A rock is propelled upward from the top of a building 60 feet tall at an initial velocity of 70 feet $p$ second. Give the function that describes the height of the rocket in terms of time $t$.
A) $f ( t ) = - 16 t ^ { 2 } + 60 t + 70$
B) $f ( t ) = - 16 t ^ { 2 } + 70 t + 60$
C) $f ( t ) = 16 t ^ { 2 } + 70 t + 60$
D) $f ( t ) = - 70 t ^ { 2 } + 16 t + 60$

Accepted Answer

The correct function that describes the height of the rock in terms of time $t$ is given by the equation of motion under gravity, which is $f(t) = -16t^2 + vt + h$, where $v$ is the initial velocity (70 feet per second) and $h$ is the initial height (60 feet). Thus, the correct equation is $f(t) = -16t^2 + 70t + 60$.

Question 9

Solve the problem.
-John owns a hotdog stand. His profit is represented by the equation $P ( x ) = - x ^ { 2 } + 12 x + 44$, with $P$ being profits and $x$ the number of hotdogs sold. What is the most he can earn?
A) $\$ 80$
B) $\$ 36$
C) $\$ 104$
D) $\$ 68$

Accepted Answer

The given equation $P(x) = -x^2 + 12x + 44$ is a quadratic equation that opens downwards (since the coefficient of $x^2$ is negative), meaning it has a maximum point. The maximum profit occurs at the vertex of the parabola. The x-coordinate of the vertex can be found using the formula $-\frac{b}{2a}$, where $a = -1$ and $b = 12$. Thus, the x-coordinate is $-\frac{12}{2(-1)} = 6$. Substituting $x = 6$ back into the profit equation gives $P(6) = -(6)^2 + 12(6) + 44 = -36 + 72 + 44 = 80$. Therefore, the maximum profit John can earn is $80.

Question 10

Solve the problem.
-A coin is tossed upward from a balcony 242 feet high with an initial velocity of 16 feet per second, then its height after $t$ seconds is given by the equation $h ( t ) = - 16 t ^ { 2 } + 16 t + 242$. During what interval of time will the coin be at a height of at least $50 \mathrm { ft }$ ?
A) $4 \leq t \leq 8$
B) $3 \leq t \leq 4$
C) $0 \leq t \leq 1$
D) $0 \leq t \leq 4$

Accepted Answer

To find the interval of time during which the coin will be at a height of at least 50 ft, we need to solve the inequality $ -16t^2 + 16t + 242 \geq 50 $. Subtracting 50 from both sides gives $ -16t^2 + 16t + 192 \geq 0 $. Factoring out -16 gives $ -16(t^2 - t - 12) \geq 0 $, which simplifies to $ t^2 - t - 12 \leq 0 $ (reversing the inequality sign because we divided by a negative number). Factoring the quadratic equation gives $ (t-4)(t+3) \leq 0 $. By the zero product property, the solutions are $ t = 4 $ and $ t = -3 $. Since time cannot be negative, we disregard $ t = -3 $. Testing intervals around $ t = 4 $, we find that the inequality holds for $ 0 \leq t \leq 4 $, indicating the coin is at least 50 ft high during this time interval.

Question 11

Solve the problem.
-Bob owns a watch repair shop. He has found that the cost of operating his shop is given by $c ( x ) = 3 x ^ { 2 } - 204 x + 46$, where $c$ is cost and $x$ is the number of watches repaired. How many watches must he repair to have the lowest cost?
A) 46 watches
B) 34 watches
C) 30 watches
D) 23 watches

Accepted Answer

The lowest cost occurs at the vertex of the parabola represented by the given quadratic equation. The x-coordinate of the vertex can be found using the formula $-\frac{b}{2a}$, where $a = 3$ and $b = -204$. Plugging these values in gives $-\frac{-204}{2*3} = 34$. Therefore, Bob must repair 34 watches to have the lowest cost.

Question 12

Solve the problem.
-A rock is propelled upward from the top of a building 160 feet tall at an initial velocity of 48 feet per second. The function that describes the height of the rocket in terms of time $t$ is $s ( t ) = - 16 t ^ { 2 } + 48 t + 160$. Determine the maximum height that the rock reaches.
A) $196 \mathrm { ft }$
B) $166 \mathrm { ft }$
C) $179 \mathrm { ft }$
D) $213 \mathrm { ft }$

Accepted Answer

The maximum height can be found by using the vertex formula for a quadratic equation $s(t) = -16t^2 + 48t + 160$. The time at which the maximum height is reached is given by $t = -\frac{b}{2a}$, where $a = -16$ and $b = 48$. Thus, $t = -\frac{48}{2(-16)} = 1.5$ seconds. Substituting $t = 1.5$ into the equation gives $s(1.5) = -16(1.5)^2 + 48(1.5) + 160 = 196$ feet.

Question 13

Solve the problem.
-A farmer has 500 feet of fence with which to fence a rectangular plot of land. The plot lies along river so that only three sides need to be fenced. Estimate the largest area that can be fenced.
A) $46,875 \mathrm { ft } ^ { 2 }$
B) $37,500 \mathrm { ft } ^ { 2 }$
C) $31,250 \mathrm { ft } ^ { 2 }$
D) $25,000 \mathrm { ft } ^ { 2 }$

Accepted Answer

The answer of Solve the problem.
-A farmer has 500 feet...

Question 14

Solve the problem.
-If an object is propelled upward from a height of 160 feet at an initial velocity of 144 feet per second, then its height after $t$ seconds is given by the equation $h ( t ) = - 16 t ^ { 2 } + 144 t + 160$. After how many seconds does the object hit the ground?
A) $11 \mathrm { sec }$
B) $5.0 \mathrm { sec }$
C) $9.0 \mathrm { sec }$
D) $10 \mathrm { sec }$

Accepted Answer

To find when the object hits the ground, set $h(t) = 0$ and solve $-16t^2 + 144t + 160 = 0$. Solving this quadratic equation gives $t = 10$ seconds.

Question 15

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 (x) weeks after its release. How many units can you expect to sell in week 12?
A) 5784 units
B) 4068 units
C) 2616 units
D) 4332 units

Accepted Answer

Substituting $x = 12$ into the quadratic model $y = -11x^2 + 350x$, we get $y = -11(12)^2 + 350(12) = -11(144) + 4200 = -1584 + 4200 = 2616$ units.

Question 16

Solve the problem.
-Of all possible pairs of positive numbers having a sum of 78 , which pair has the maximum product?
A) 30 and 48
B) 39 and 39
C) 38 and 40
D) 29 and 49

Accepted Answer

The maximum product of two numbers with a fixed sum occurs when the two numbers are equal, in this case, 78/2 = 39.

Question 17

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 $s ( t ) = - 16 \left( t ^ { 2 } \right) + 178 t$. At what time(s) will the ball be $235 \mathrm { ft }$ from the ground? Round your answer to the nearest tenth of a second.
A) $5.6 \mathrm { sec }$
B) $11.1 \mathrm { sec }$
C) $172.8 \mathrm { sec }$ and $183.2 \mathrm { sec }$
D) $1.5 \mathrm { sec }$ and $9.6 \mathrm { sec }$

Accepted Answer

To find when the rocket is 235 ft from the ground, solve the equation $-16t^2 + 178t = 235$. Rearrange to $-16t^2 + 178t - 235 = 0$ and use the quadratic formula to find $t$. The solutions are approximately $t = 1.5$ and $t = 9.6$ seconds.

Question 18

Solve the problem.
-A rock is propelled upward from the top of a building 230 feet tall at an initial velocity of 192 feet per second. The function that describes the height of the rocket in terms of time $t$ is $s ( t ) = - 16 t ^ { 2 } + 192 t + 230$. Determine the time at which the rock reaches its maximum height.
A) $9.6 \mathrm { sec }$
B) $4.8 \mathrm { sec }$
C) $12 \mathrm { sec }$
D) $6 \mathrm { sec }$

Accepted Answer

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

Question 19

Solve the problem.
-John owns a hot dog stand. He has found that his profit is represented by the equation $P ( x ) = - x ^ { 2 } + 56 x + 71$, with $P$ being profits and $x$ the number of hot dogs sold. How many hot dogs must he sell to earn the most profit?
A) 43 hot dogs
B) 21 hot dogs
C) 29 hot dogs
D) 28 hot dogs

Accepted Answer

The equation $P(x) = -x^2 + 56x + 71$ 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 number of hot dogs sold for maximum profit. Here, $a = -1$, $b = 56$, so $x = -\frac{56}{2(-1)} = 28$. Therefore, John must sell 28 hot dogs to earn the most profit.

Question 20

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 77 feet?
A) $1.2 \mathrm { sec }$
B) $2.0 \mathrm { sec }$
C) $2.2 \mathrm { sec }$
D) $1.4 \mathrm { sec }$

Accepted Answer

To find the time it takes for the ball to fall 77 feet, set $s(t) = 77$ and solve the equation $16t^2 + 32t = 77$. Solving the quadratic equation gives $t \approx 1.4$ seconds.

Quiz 3: Polynomial and Rational Functions

Solve the problem. - $x$ and $y$ are two positive numbers and $y$ is three greater than $x$ . The product of the numbers is 270 . Find the smaller number, $x$ .

Solve the problem. -The length of a table is 11 inches more than its width. If the area of the table is 1692 square inches, what is its length?

Solve the problem. -If a rocket is propelled upward from ground level, its height in meters after $t$ seconds is given by $h ( t ) = - 9.8 t ^ { 2 } + 98 t$ . During what interval of time will the rocket be higher than $235.2 \mathrm {~m}$ ?

Solve the problem. -The number of mosquitoes $M ( x )$ , in millions, in a certain area depends on the June rainfall $x$ , in inches, according to the equation $M ( x ) = 6 x - x ^ { 2 }$ . What rainfall produces the maximum number of mosquitoes?

Solve the problem. -A rock is propelled upward from the top of a building 60 feet tall at an initial velocity of 70 feet $p$ second. Give the function that describes the height of the rocket in terms of time $t$ .

Solve the problem. -John owns a hotdog stand. His profit is represented by the equation $P ( x ) = - x ^ { 2 } + 12 x + 44$ , with $P$ being profits and $x$ the number of hotdogs sold. What is the most he can earn?

Solve the problem. -Bob owns a watch repair shop. He has found that the cost of operating his shop is given by $c ( x ) = 3 x ^ { 2 } - 204 x + 46$ , 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. -A farmer has 500 feet of fence with which to fence a rectangular plot of land. The plot lies along river so that only three sides need to be fenced. Estimate the largest area that can be fenced.

Solve the problem. -If an object is propelled upward from a height of 160 feet at an initial velocity of 144 feet per second, then its height after $t$ seconds is given by the equation $h ( t ) = - 16 t ^ { 2 } + 144 t + 160$ . After how many seconds does the object hit the ground?

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 (x) weeks after its release. How many units can you expect to sell in week 12?

Solve the problem. -Of all possible pairs of positive numbers having a sum of 78 , which pair has the maximum product?

Solve the problem. -John owns a hot dog stand. He has found that his profit is represented by the equation $P ( x ) = - x ^ { 2 } + 56 x + 71$ , with $P$ being profits and $x$ the number of hot dogs sold. How many hot dogs must he sell to earn the most profit?

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 77 feet?

Quiz 3: Polynomial and Rational Functions

Solve the problem. - xxx and yyy are two positive numbers and yyy is three greater than xxx . The product of the numbers is 270 . Find the smaller number, xxx .

Solve the problem. -The length of a table is 11 inches more than its width. If the area of the table is 1692 square inches, what is its length?

Solve the problem. -If a rocket is propelled upward from ground level, its height in meters after ttt seconds is given by h(t) =−9.8t2+98th ( t ) = - 9.8 t ^ { 2 } + 98 th(t) =−9.8t2+98t . During what interval of time will the rocket be higher than 235.2 m235.2 \mathrm {~m}235.2 m ?

Solve the problem. -The number of mosquitoes M(x) M ( x ) M(x) , in millions, in a certain area depends on the June rainfall xxx , in inches, according to the equation M(x) =6x−x2M ( x ) = 6 x - x ^ { 2 }M(x) =6x−x2 . What rainfall produces the maximum number of mosquitoes?

Solve the problem. -A rock is propelled upward from the top of a building 60 feet tall at an initial velocity of 70 feet ppp second. Give the function that describes the height of the rocket in terms of time ttt .

Solve the problem. -John owns a hotdog stand. His profit is represented by the equation P(x) =−x2+12x+44P ( x ) = - x ^ { 2 } + 12 x + 44P(x) =−x2+12x+44 , with PPP being profits and xxx the number of hotdogs sold. What is the most he can earn?