Quixote wants to fence in his rectangular garden and has $6000 to spend. One side (one of the x sides/width sides) needs a stronger fence to keep the cows out--materials will cost $10 per foot for this side. The other three sides can be fenced with less expensive materials at $5 per foot.
Write an equation using x (width) and y (length) to express Quixote's cost to build this rectangular fence:
6000 = ?
Solve the above equation for y and use substitution to find an equation to express area of the garden as a function of x :
A(x) = ?
Use your area equation to find the dimensions which will maximize the area enclosed in the garden.
x= ?
y= ?
Write an equation using x (width) and y (length) to express Quixote's cost to build this rectangular fence:
6000 = ?
Solve the above equation for y and use substitution to find an equation to express area of the garden as a function of x :
A(x) = ?
Use your area equation to find the dimensions which will maximize the area enclosed in the garden.
x= ?
y= ?
-
10*x + 5*(x + 2*y) = 6000 => 15*x + 10*y = 6000
=> y = 600 - 1.5*x..................(1)
A = x*y = 600*x - 1.5*x^2.
For maximum A'=0, A''<0 =>
A' = 600 - 3x = 0 => x = 200 => from (1) y = 300.
Verify: A'' = -3 < 0 hence maximum.
=> y = 600 - 1.5*x..................(1)
A = x*y = 600*x - 1.5*x^2.
For maximum A'=0, A''<0 =>
A' = 600 - 3x = 0 => x = 200 => from (1) y = 300.
Verify: A'' = -3 < 0 hence maximum.