Find the dimensions of the rectangular corral split into 2 p

Find the dimensions of the rectangular corral split into 2 pens of the same size producing the greatest possible enclosed area given 300 feet of fencing. (Assume that the length is greater than or equal to the width.)

Length____ft

Width____ft

Solution

So we have 3 sections of fencing with length x,
and 4 sections of fencing with length y

3x + 4y = 300
4y = 300 3x
y = (3/4)(100 x )

Total area = x * y

We can express area as a function of x if we substitute y with(3/4)( 100x)
A(x) = x (3/4) (100x)
A(x) = 75x -(3/4)x^2

A\'(x) = 75 (3/2) x = 0
75x = 3/2
x = 50 ft (length)

y = (3/4)(100 50) = 150/4 = 37.5 ft (width)

Dimensions that produce greatest enclosed area = 50 ft * 37.5 ft = 1875 ft².

Mathmom · 4 years ago