Question

Could I please get some help with the following question:

The Smith's own a home on the water. The area to be fenced is 320,000 square feet. Use
calculus to determine the minimum amount of fencing that must be used to enclose the remaining three
sides of the property.

I'd really appreciate any help I can get.

Answer 1

You left some things out when writing the problem. Like the first side that is of a given length...

Answer 2

@knivez that was all the info I was given.

Answer 3

Area = x * y = 320,000 (1)

Perimeter P = x + y + y
P = x + 2y (2)

From (1): x = 320000/y. Put this in (2):
P = 320000/y + 2y

For minimum fence length, dP/dy = 0

Find dP/dy, set it to zero and solve for y. Then find x and P.

Answer 4

oh I guess the original side is not needed to maximize the remaining three. Ranga made a very useful diagram. From his diagram you should be able to make 2 equations with 2 variables each and solve.

Answer 5

I'm not sure how to find dp/dy. Could you help me bit more please?

Answer 6

would dP/dy be (-320,000/y^2) + 2

Answer 7

P = 320000/y + 2y
P = 320000 * y^(-1) + 2y
Differentiate with respect to y:
dP/dy = (-1)320000 * y^(-2) + 2 = 0
-320000/y^2 + 2 = 0
2 = 320000/y^2
y^2 = 320000/2 = 160000 = 16 * 10^4
y = 4 * 10^2 = 400

Answer 8

From (1) x = 320,000 / y = 320,000 / 400 = 800
From (2) P = x + 2y = 800 + (2 * 400) = 800 + 800 = 1600 feet.

Answer 9

GREAT! thankyou so much!!

Answer 10

you are welcome.