A construction company wishes to build a rectangular enclosure to store machiner

Question

A construction company wishes to build a rectangular enclosure to store machinery and equipment. The site selected borders on a river that will be used as one of the sides of the rectangle. Fencing will be needed to form the other three sides. The company has 528 feet of 10- foot high chain-link fencing. The questions that follow should help you determine the dimensions of the rectangle that will produce the maximum area for the storage site. Assume that the company wishes to use all of the fencing, and that “width” refers to the measure of each of the two sides that are perpendicular to the river, and “length” refers to the measure of the side parallel to the river (see diagram). For the questions that follow, all measurement answers are to be given with the correct unit label, e.g. feet, square feet, etc. 1. Copy the following table into your project and complete it, showing the dimensions (width and length) of some possible enclosures along with the resulting areas. (Recall that the company is using 528 feet of fencing total.) The table is Width(feet) 25 50 75 100 125 150 175 200 Length(feet)--------------------------------------------- Area(sq.Ft----------------------------------------------- Need to have length and area filled in . Of all the dimensions (length and width) listed in the table, which choice gives the largest area for the enclosure? Express your answer with appropriate units. 3. Let w represent the width, and l represent the length. Write an equation expressing l in terms of w. 4. Write the general formula for the area A of a rectangle in terms of l and w. 5. Use the general formula to write the area A of the enclosure as a function of w (that is, only the variable w should appear in the function). Write the function in simplest form. What type of function is this? 7. What do we call the graph of this area function? 8. What is the name of the point that represents the maximum area? 9. Find the width needed to give the maximum area algebraically. Show the formula you used and the steps to the solution. Express your answer with appropriate unit. 10. Now that you know the width needed to find the maximum area, find the corresponding length of the enclosure of maximum area using the formula from question 3. Show the steps to your solution. Express your answer with appropriate unit. 11. Now find the maximum area algebraically using the function A(w) . Express your answer with appropriate unit.

Explanation / Answer

let l be length

w be the width

Area = l*w

Total fence = 528 feet

Perimeter of fence = 2w +l

2w +l = 528----> l = 528-2w

Area = (528-2w)w = - 2w^2 + 528w

Its a quadratice function

The maximium area occurs at the vertex---w= -b/2a = -528/-2*2 = 132 feet

width = 132 feet;

length = 528- 2*132 = 528 - 264 = 264 feet

Corresponding maximium area = 132*264 = 34848 feet^2

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