Question 232503
A rancher needs to enclose two adjacent rectangular corrals, one for cattle and one for sheep. If the river forms one side of the corrals and 330 yd of fencing is available, what is the largest total area that can be enclosed?
:
Let W = the three sides to make the width of the two corrals
Let L = the one side parallel to the river
:
L + 3W = 330 yds (of fence available)
L = (330-3W); this form for substitution
;
Area
A = L * W
Replace L with (330-3W)
A = (330-3W) * W
A = -3W^2 + 330W
A quadratic equation, the axis of symmetry will be the value for max area
Find that using x = -b/(2a)
In this equations: x = W; a = -3; b = 330
W = {{{(-330)/(2*-3)}}}
:
W = {{{(-330)/(-6)}}}
W = +55 yd is the width for max area
:
Find the max area, substitute 55 for W in the area equation:
A = -3(55^2) + 330(55)
A = -3(3025) + 18150
A = -9075 + 18150
A = +9075 sq/yds is max area
;
:
Check this; Find the length
L = 330 - 3(55)
L = 330 - 165
L = 165 yds
;
Find the area: 165 * 55 = 9075 sq/yds