Question

The fencing of the left border costs $4 per foot, while the fencing of the lower border costs $1 per foot. (no fencing is required along the river.) you want to spend $64 and enclose as much area as possible. what are the dimensions of your garden, and what area does it enclose? hint [the area of a right-triangle is given by

Answer 1

Answer
Left border = 8 feet
Lower border =32 feet
Area=128 〖ft〗^2


Explanation
The area is a right triangle.
Let us take the left border to be y feet and the lower border to be x feet
Area=1/2 yx
The cost of fencing would be 4y+x=64.
From this expression x=64-4y
Substituting this value of x in the area we have;
Area=1/2 y(64-4y)
A=32y-2y^2
To get the maximum area of the triangle the first derivative of the area would te equal to zero.
A=32y-2y^2
dA/dy=32-4y=0
4y=32
y=8
Since x=64-4y
x=64-(4×8)=32
The dimensions of the garden are 8 feet by 32 feet.
Since the garden is a right triangle,
Area=1/2×8×32
Area=128 〖ft〗^2