Quadratic Functions

Question

Elli has 1400m of fencing and wants to fence off a rectangle ostrich run that is at the bottom of a cliff face. She needs no fence along the cliff face. what are the dimensions of the ostrich run that has the largest area?

Answer 1

Hello, Jordan,

I'll use A (area), L (Length), W (Width) to make the following equations:

A = W*L

Normally, the perimeter of a rectangular fence would be 2L + 2W, but in this case the perimeter is 1L + 2W,

since the other side of the portion defined by L is the cliff, where no fencing is needed. Therefore the fencing length is defined by:

1400ft = L + 2W

Rewriting this: L = 1400 - 2W

Area is therefore:

A = W*(1400 - 2W), or =1400W - 2W²

Rewriting this in terms of x and y: y = -2x² +1400x, which is a parabola. Plot, or differentiate to find the maximum point, which is 350 ft. An rectangular pen with maximum area with 1400 ft of fencing would be 350ft long (parallel to the cliff) and (1400 - 350)/2 = 525 feet wide, for an area of 350*525 = 183,750 ft².

Bob

Answer 2

L+2W = 1400

Area = A = LW = (1400-2W)W = 1400W- 2W^2

A' = 1400-4W =0

4W = 1400

W = 350

L= 1400-700 = 700

750m by 350 m for max area