Question 1

Solve the problem
-A rain gutter is made from sheets of aluminum that are 18 inches wide by turning up the edges to form right angles. Determine the depth of the gutter that will maximize its cross-sectional area and allow the greatest Amount of water to flow.

Accepted Answer

The cross-sectional area of the gutter is maximized when the width of the base and the height of the sides are equal. If the total width is 18 inches, then the base plus two sides equals 18 inches. Let x be the height of the sides, then the base is (18 - 2x). The area A = x(18 - 2x). To maximize A, take the derivative and set it to zero: A' = 18 - 4x = 0, solving gives x = 4.5 inches.

Question 2

Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of
the minimum or maximum point.
-$$f ( x ) = 4 x ^ { 2 } - 12 x$$

Accepted Answer

The quadratic function $f(x) = 4x^2 - 12x$ opens upwards because the coefficient of $x^2$ is positive, indicating a minimum value. The vertex (minimum point) can be found using the formula $x = -\frac{b}{2a}$, giving $x = \frac{3}{2}$. Substituting $x = \frac{3}{2}$ into the function gives the minimum value $f\left(\frac{3}{2}\right) = -9$, so the correct answer is minimum; $\left(\frac{3}{2}, -9\right)$.

Question 3

Solve the problem
-Among all pairs of numbers whose difference is 36, find a pair whose product is as small as possible.

Accepted Answer

The product of two numbers with a fixed difference is minimized when the numbers are equidistant from zero, hence -18 and 18.

Question 4

Solve the problem
-You have 64 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area.

Accepted Answer

The maximum area of a rectangle given a fixed perimeter is achieved when the rectangle is a square. However, since one side (the side along the river) does not require fencing, the problem effectively involves only three sides. To maximize the area, the two widths and one length should be equal, making the two widths (sides perpendicular to the river) and the length (parallel to the river) use the fencing efficiently. With 64 feet of fencing and not fencing one side along the river, the best allocation is for the two widths to total 32 feet (16 feet each) and the length parallel to the river to be 32 feet, making the correct answer length: 32 feet, width: 16 feet.

Question 5

Solve the problem
-The manufacturer of a CD player has found that the revenue $R$ (in dollars) is $R ( p ) = - 4 p ^ { 2 } + 1560 p$, when the unit price is $p$ dollars. If the manufacturer sets the price $p$ to maximize revenue, what is the maximum revenue to the nearest whole dollar?

Accepted Answer

The maximum revenue can be found by completing the square or using the vertex formula for a parabola. The revenue function is a quadratic equation in the form of $R(p) = -4p^2 + 1560p$. The vertex of this parabola, which gives the maximum point, occurs at $p = -\frac{b}{2a}$, where $a = -4$ and $b = 1560$. Thus, $p = -\frac{1560}{2(-4)} = 195$. Substituting $p = 195$ into the revenue function gives the maximum revenue: $R(195) = -4(195)^2 + 1560(195) = 152,100$.

Question 6

Solve the problem
-The daily profit in dollars of a specialty cake shop is described by the function $P ( x ) = - 4 x ^ { 2 } + 208 x - 1920$, where $x$ is the number of cakes prepared in one day. The maximum profit for the company occurs at the vertex of the parabola. How many cakes should be prepared per day in order to maximize profit?

Accepted Answer

The vertex of a parabola given by $P(x) = ax^2 + bx + c$ can be found using the formula $x = -\frac{b}{2a}$. Here, $a = -4$ and $b = 208$, so $x = -\frac{208}{2(-4)} = 26$. This means 26 cakes should be prepared per day to maximize profit.

Question 7

Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of
the minimum or maximum point.
-$$f ( x ) = 2 x ^ { 2 } - 2 x - 3$$

Accepted Answer

The quadratic function $f(x) = 2x^2 - 2x - 3$ has a positive leading coefficient (2), indicating it opens upwards and thus has a minimum value. The vertex (minimum point) of a parabola given by $f(x) = ax^2 + bx + c$ can be found using the formula $x = -\frac{b}{2a}$. Here, $a = 2$ and $b = -2$, so $x = -\frac{-2}{2*2} = \frac{1}{2}$. Substituting $x = \frac{1}{2}$ into the original equation gives $f\left(\frac{1}{2}\right) = 2\left(\frac{1}{2}\right)^2 - 2\left(\frac{1}{2}\right) - 3 = -\frac{7}{2}$. Therefore, the minimum point is $\left(\frac{1}{2}, -\frac{7}{2}\right)$.

Question 8

Solve the problem
-In one U.S. city, the quadratic function $f ( x ) = 0.0042 x ^ { 2 } - 0.46 x + 36.44$ models the median, or average, age, $y$, at which men were first married x years after 1900 . In which year was this average age at a minimum? (Round to the nearest year.) What was the average age at first marriage for that year? (Round to the nearest tenth.)

Accepted Answer

To find the year when the average age at first marriage was at a minimum, we need to find the vertex of the quadratic function. The x-coordinate of the vertex can be found using the formula $-\frac{b}{2a}$, where $a = 0.0042$ and $b = -0.46$. Plugging these values in, we get $-\frac{-0.46}{2(0.0042)} = 54.76$. Since $x$ represents years after 1900, we add this to 1900 to get 1954.76, which rounds to 1955. To find the average age at that year, we substitute $x = 54.76$ into the original equation to get $f(54.76) = 0.0042(54.76)^2 - 0.46(54.76) + 36.44$, which approximately equals 23.8. Therefore, the correct answer is 1955, with an average age of 23.8 years old.

Question 9

Solve the problem
-A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. 600 feet of fencing is used. Find the maximum area of the playground.

Accepted Answer

The answer of Solve the problem
-A rectangular playground is to...

Question 10

Solve the problem
-Among all pairs of numbers whose sum is 68, find a pair whose product is as large as possible.

Accepted Answer

The product of two numbers is maximized when the numbers are as close to each other as possible, given a fixed sum. Here, the sum is 68, so the numbers that are closest to each other and sum up to 68 are 34 and 34.

Question 11

Solve the problem
-The owner of a video store has determined that the profits $\mathrm { P }$ of the store are approximately given by $\mathrm { P } ( \mathrm { x } ) = - \mathrm { x } ^ { 2 } + 80 \mathrm { x } + 72$, where $\mathrm { x }$ is the number of videos rented daily. Find the maximum profit to the nearest dollar.

Accepted Answer

The answer of Solve the problem
-The owner of a video...

Question 12

Solve the problem
-You have 120 feet of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area.

Accepted Answer

To maximize the area of a rectangle with a fixed perimeter, the rectangle should be a square. With a perimeter of 120 feet, each side of the square would be 30 feet (since 4 sides of 30 feet each equals 120 feet). Thus, the dimensions are 30 ft by 30 ft.

Question 13

Solve the problem
-A developer wants to enclose a rectangular grassy lot that borders a city street for parking. If the developer has 332 feet of fencing and does not fence the side along the street, what is the largest area that can be enclosed?

Accepted Answer

The largest area for a given perimeter is achieved with a square, but since one side is open, we treat it as a rectangle where one pair of opposite sides (the width) is equal, and the length is twice the width. Let the width be $w$ and the length be $2w$. The total fencing used is the perimeter of the three sides, $2w + 2w + w = 332$, simplifying to $5w = 332$, so $w = 66.4$ feet. The area is $w \times 2w = 66.4 \times 132.8 = 13,778 \mathrm { ft } ^ { 2 }$.

Question 14

Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of
the minimum or maximum point.
-$$f ( x ) = - 2 x ^ { 2 } + 6 x$$

Accepted Answer

The coefficient of $x^2$ in the quadratic function $f(x) = -2x^2 + 6x$ is negative, indicating that the parabola opens downwards and thus has a maximum value. To find the x-coordinate of the vertex (maximum point), use the formula $-\frac{b}{2a}$, where $a = -2$ and $b = 6$. This gives $-\frac{6}{2(-2)} = \frac{3}{2}$. Substituting $x = \frac{3}{2}$ into the function gives $f\left(\frac{3}{2}\right) = -2\left(\frac{3}{2}\right)^2 + 6\left(\frac{3}{2}\right) = -2\left(\frac{9}{4}\right) + 9 = -\frac{9}{2} + 9 = \frac{9}{2}$. Therefore, the maximum point is $\left(\frac{3}{2}, \frac{9}{2}\right)$.

Question 15

Solve the problem
-An arrow is fired into the air with an initial velocity of 96 feet per second. The height in feet of the arrow t seconds after it was shot into the air is given by the function $h ( x ) = - 16 t ^ { 2 } + 96 t$. Find the maximum height of the arrow.

Accepted Answer

The maximum height of the arrow is found by determining the vertex of the parabola described by the function $h(t) = -16t^2 + 96t$. The vertex occurs at $t = -\frac{b}{2a}$, where $a = -16$ and $b = 96$. This gives $t = 3$. Substituting $t = 3$ into the function gives $h(3) = -16(3)^2 + 96(3) = 144$ feet.

Question 16

Solve the problem
-You have 240 feet of fencing to enclose a rectangular region. What is the maximum area?

Accepted Answer

The maximum area that can be enclosed by a given length of fencing is achieved by forming a square. The perimeter of the square is 240 feet, so each side is 240/4 = 60 feet. The area of the square is 60^2 = 3600 square feet.

Question 17

Solve the problem
-A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. 600 feet of fencing is used. Find the dimensions of the playground that maximize the total Enclosed area.

Accepted Answer

To maximize the area, let the length be x and the width be y. The total fencing used is 2x + 3y = 600. Solving for y gives y = (600 - 2x)/3. The area A = x*y = x(600 - 2x)/3. To maximize A, take the derivative and set it to zero. Solving gives x = 150, y = 100, so the dimensions are 100 ft by 150 ft.

Question 18

Solve the problem
-The cost in millions of dollars for a company to manufacture $x$ thousand automobiles is given by the function $C ( x ) = 4 x ^ { 2 } - 32 x + 128$. Find the number of automobiles that must be produced to minimize the cost.

Accepted Answer

The answer of Solve the problem
-The cost in millions of...

Question 19

Solve the problem
-The profit that the vendor makes per day by selling $x$ pretzels is given by the function $P ( x ) = - 0.004 x ^ { 2 } + 2.4 x - 400$. Find the number of pretzels that must be sold to maximize profit.

Accepted Answer

The profit function $P(x) = -0.004x^2 + 2.4x - 400$ 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 number of pretzels that maximizes profit occurs at the vertex of the parabola. The x-coordinate of the vertex can be found using the formula $-\frac{b}{2a}$. Here, $a = -0.004$ and $b = 2.4$, so $-\frac{2.4}{2(-0.004)} = 300$. Therefore, to maximize profit, the vendor must sell 300 pretzels.

Question 20

Solve the problem
-The owner of a video store has determined that the cost $C$, in dollars, of operating the store is approximately given by $C ( x ) = 2 x ^ { 2 } - 28 x + 740$, where $x$ is the number of videos rented daily. Find the lowest cost to the nearest dollar.

Accepted Answer

The answer of Solve the problem
-The owner of a video...

Quiz 3: Polynomial and Rational Functions

Solve the problem -A rain gutter is made from sheets of aluminum that are 18 inches wide by turning up the edges to form right angles. Determine the depth of the gutter that will maximize its cross-sectional area and allow the greatest Amount of water to flow.

Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of the minimum or maximum point. - $f ( x ) = 4 x ^ { 2 } - 12 x$

Solve the problem -Among all pairs of numbers whose difference is 36, find a pair whose product is as small as possible.

Solve the problem -You have 64 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area.

Solve the problem -The manufacturer of a CD player has found that the revenue $R$ (in dollars) is $R ( p ) = - 4 p ^ { 2 } + 1560 p$ , when the unit price is $p$ dollars. If the manufacturer sets the price $p$ to maximize revenue, what is the maximum revenue to the nearest whole dollar?

Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of the minimum or maximum point. - $f ( x ) = 2 x ^ { 2 } - 2 x - 3$

Solve the problem -A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. 600 feet of fencing is used. Find the maximum area of the playground.

Solve the problem -Among all pairs of numbers whose sum is 68, find a pair whose product is as large as possible.

Solve the problem -You have 120 feet of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area.

Solve the problem -A developer wants to enclose a rectangular grassy lot that borders a city street for parking. If the developer has 332 feet of fencing and does not fence the side along the street, what is the largest area that can be enclosed?

Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of the minimum or maximum point. - $f ( x ) = - 2 x ^ { 2 } + 6 x$

Solve the problem -An arrow is fired into the air with an initial velocity of 96 feet per second. The height in feet of the arrow t seconds after it was shot into the air is given by the function $h ( x ) = - 16 t ^ { 2 } + 96 t$ . Find the maximum height of the arrow.

Solve the problem -You have 240 feet of fencing to enclose a rectangular region. What is the maximum area?

Solve the problem -A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. 600 feet of fencing is used. Find the dimensions of the playground that maximize the total Enclosed area.

Solve the problem -The cost in millions of dollars for a company to manufacture $x$ thousand automobiles is given by the function $C ( x ) = 4 x ^ { 2 } - 32 x + 128$ . Find the number of automobiles that must be produced to minimize the cost.

Solve the problem -The profit that the vendor makes per day by selling $x$ pretzels is given by the function $P ( x ) = - 0.004 x ^ { 2 } + 2.4 x - 400$ . Find the number of pretzels that must be sold to maximize profit.

Solve the problem -The owner of a video store has determined that the cost $C$ , in dollars, of operating the store is approximately given by $C ( x ) = 2 x ^ { 2 } - 28 x + 740$ , where $x$ is the number of videos rented daily. Find the lowest cost to the nearest dollar.