Mr. Rogers wants to fence in a rectangular garden next to a straight section of the Scioto

Question

River. He has 330 feet of fencing to do the entire job. He doesn’t need fencing along the
river, and there is a 4.5 foot wide clearance for a gate at one end.
What are the dimensions of the garden (x and y) with the largest area.

Thanks!

Answer 1

let the width be x
and the length be y
so we are given ....
2x - 4.5 + y = 330
y = 334.5 - 2x

area = xy
= x(334.5-2x)
= 334.5x - 2x^2

d(area)/dx = 334.5 - 4x = 0 for a max area
4x = 334.5
x = 334.5/4 = 83.625
y = 334.5 - 2(83.625) = 167.25

check:
167.25+83.625+83.625-4.5 = 330
area = xy = 13986.28125

let x = 83 , then y = 168.5
area = 13985.5 , a bit smaller

let x = 84 , then y = 166.5
area = xy = 13986 , also a bit smaller than the above
answer looks good.