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Maximize/Minimize Quadratic Functions Help (page 3)

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By — McGraw-Hill Professional
Updated on Oct 4, 2011

Example

  • A farmer has 1000 feet of fencing materials available to fence a rectangular pasture next to a river. If the side along the river does not need to be fenced, what dimensions will maximize the enclosed area? What is the maximum enclosed area?

    Quadratic Functions Example

    Fig. 6.11

    Using the fact that 2 W + L = 1000, we can solve for L and substitute this quantity in the area formula A = LW .

    2 W + L = 1000

    L = 1000 − 2 W

    A = LW

    A = (1000 − 2 W ) W = −2 W 2 + 1000 W

    This quadratic function has a maximum value.

    Quadratic Functions Example

    Maximize the enclosed area by letting W = 250 feet and L = 1000 − 2(250) = 500 feet. The maximum enclosed area is 125,000 square feet.

In the last problems, we will maximize the area of a figure but will have to work a little harder to find the area function to maximize.

Examples

  • A window is to be constructed in the shape of a rectangle surmounted by a semicircle (see Figure 6.12). The perimeter of the window needs to be 18 feet. What dimensions will admit the greatest amount of light?

    Quadratic Functions Examples

    Fig. 6.12

    The dimensions that will admit the greatest amount of light are the same that will maximize the area of the window. The area of the window is the rectangular area plus the area of the semicircle. The area of the rectangular region is LW . Because the width of the window is the diameter (or twice the radius) of the semicircle, we can rewrite the area as L (2 r ) = 2 rL. The area of the semicircle is half of the area of a circle with radius r, Quadratic Functions Examples. The total area of the window is

    Quadratic Functions Examples

    Now we will use the fact that the perimeter is 18 feet to help us replace L with an expression using r . The perimeter is made up of the two sides (2L) and the bottom of the rectangle (2 r ) and the length around the semicircle. The length around the outside of the semicircle is half of the circumference of a circle with radius r , Quadratic Functions Examples . The total perimeter is P = 2 L + 2 r + π r . This is equal to 18. We will solve the equation 2 L + 2 r + π r = 18 for L.

    Quadratic Functions Examples

    Now we will substitute Quadratic Functions Examples for L in the area formula.

    Quadratic Functions Examples

     

    Quadratic Functions Examples

    This quadratic function has a maximum value.

    Quadratic Functions Examples

    Maximize the amount of light admitted in the window by letting the radius of the semicircle be about 2.52 feet, and the length about Quadratic Functions Examples 2.52 feet.

  • A track is to be constructed so that it is shaped like Figure 6.13, a rectangle with a semicircle at each end. If the inside perimeter of the track is to be Quadratic Functions Examples mile, what is the maximum area of the rectangle?

    Quadratic Functions Examples

    Fig. 6.13

    The length of the rectangle is L . Its width is the diameter of the semicircles (or twice their radius). The area formula for the rectangle is A = LW = L (2 r ) = 2 rL . The perimeter of the figure is the two sides of the rectangle (2 L ) plus the length around each semicircle (π r ). The total perimeter is 2 L + 2π r . Although we could work with the dimensions in miles, it will be easier to convert 1/4 mile to feet. There are 5280/4 = 1320 feet in 1/4 mile.

    We will solve 2 L + 2π r = 1320 for L . Solving for r works, too.

    Quadratic Functions Examples

    The area function has a maximum value.

    Quadratic Functions Examples

    The maximum area of the rectangular region is about 69,328 square feet.

  • A rectangle is to be constructed so that it is bounded below by the x -axis, on the left by the y -axis, and above by the line y = −2 x + 12. (See Figure 6.14). What is the maximum area of the rectangle?

    Quadratic Functions Examples

    Fig. 6.14

    The coordinates of the corners will help us to see how we can find the length and width of the rectangle.

    Quadratic Functions Examples

    Fig. 6.15

    The height of the rectangle is y and the width is x . This makes the area A = xy . We need to eliminate x or y . Because y = −2 x + 12, we can substitute − 2 x + 12 for y in A = xy to make it the quadratic function A = xy = x (−2 x + 12) = −2 x 2 + 12 x .

    Quadratic Functions Examples

    The maximum area is 18 square units.

Maximize/Minimize Quadratic Functions Practice Problems

Practice

  1. The average cost of a product can be approximated by the function C ( x ) = 0.00025 x 2 − 0.25 x + 70.5, where x is the number of units produced and C ( x ) is the average cost in dollars. What level of production will minimize the average cost?
  2. A frog jumps from a rock to the shore of a pond. Its path is given by the equation Quadratic Functions Practice , where x is the horizontal distance in inches, and y is the height in inches. What is the frog’s maximum height? How far had it traveled horizontally when it reached its maximum height?
  3. A projectile is fired upward from a ten-foot platform. The projectile’s initial velocity is 108 miles per hour. What is the projectile’s maximum height? When will it reach its maximum height?
  4. Attendance at home games for a college basketball team averages 1000 and the ticket price is $12. Concession sales average $2 per person. A student survey reveals that for every $0.25 decrease in the ticket price, 25 more students will attend the home games. What ticket price will maximize revenue? What is the maximum revenue?
  5. A school has 1600 feet of fencing available to enclose three playing fields (see Figure 6.16). What dimensions will maximize the enclosed area?

    Quadratic Functions Practice

    Fig. 6.16

  6. The manager of a large warehouse wants to enclose an area behind the building. He has 900 feet of fencing available. What dimensions will maximize the enclosed area? What is the maximum area?

    Quadratic Functions Practice

    Fig. 6.17

  7. A swimming pool is to be constructed in the shape of a rectangle with a semicircle at one end (see Figure 6.12). If the perimeter is to be 120 feet, what dimensions will maximize the area? What is the maximum area?

  8. A rectangle is to be constructed so that it is bounded by the x -axis, the y -axis, and the line y = −3 x + 4 (see Figure 6.18). What is the maximum area of the rectangle?

    Quadratic Functions Practice

    Fig. 6.18

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