Question

yler and Bethany have 240 feet of fencing with which to build a garden. Tyler wants the garden to be in the shape of a square, while Bethany wants it to be rectangular with a length of 50 feet and a width of 70 feet. Which design would give the maximum area for the garden? Explain. Bethany’s design would give the larger garden because the area would be 3,500 ft2. Tyler’s design would give the larger garden because the area would be 3,600 ft2. Bethany’s and Tyler’s designs would give a garden with the same area. Bethany’s and Tyler’s designs are not possible with the given amount of fencing.

Answer 1

To get which design would have maximum area we need to evaluate the area for Tyler's design. Given that the design is square, let the length= xft,  width=(120-x)
thus:
area will be:
P(x)=x(120-x)
P(x)=120x-x²
For maximum area P'(x)=0
P'(x)=120-2x=0
thus
x=60 ft
thus for maximum area x=60 ft
thus the area will be:
Area=60×60=3600 ft²
Thus we conclude that Tyler's design is the largest. Thus:
 the answer is:
Tyler’s design would give the larger garden because the area would be 3,600 ft2.