Question

A rancher who wishes to fence off a rectangular area finds that the fencing in the east-west direction will require extra reinforcement due to strong prevailing winds. Fencing in the east-west direction will therefore cost $15 per (linear) yard, as opposed to a cost of $10 per yard for fencing in the north-south direction. The rancher wants to spend $7200 on fencing, express the area of the rectangle as a function of its width x. (by width we mean the measure in yards of a side running in the east-west direction.) Also, find Domain and find which width x yields the rectangle of largest area.

Answer 1

width=x

length = y

Answer 2

so the total cost = 2 ( 15x + 10y ) =7200

Answer 3

constraint ^

Answer 4

then A= xy

Answer 5

solve the constraint for y

y= (7200 - 30x ) / 20

Answer 6

sub into the area function

Answer 7

A= (1/30 ) ( 7200x - 30x^2 )

Answer 8

dA / dx = (1/30 ) (7200 -60x ) =0

Answer 9

x=120

Answer 10

d^2 A / dx^2 = -2 <0 , so the graph is concave down and this is a maximum

Answer 11

whoops a few lines back it should be A= (1/20) ( 7200x -30x^2 ) , but that doesnt change the value of x where it is a maximum

Answer 12

whoops a few lines back it should be A= (1/20) ( 7200x -30x^2 ) , but that doesnt change the value of x where it is a maximum

Answer 13

and the second derivative is still negative