Question

A person plans to use 90 feet of fencing and the side of his house to enclose a rectangular garden. What dimensions of the rectangle would give the maximum area?

Answer 1

Draw a picture. These problems are often difficult to consider until you have a proper visualization.

Answer 2

I understand that:

so the perimeter is

90 = 2x + y

the area is

A = xy

rewrite the perimeter with y as the subject

90 - 2x = y

now substitute into the area formula

A = x(90 - 2x)

or

A = -2x^2 + 90x

the parabola is concave down so you will have a maximum

find the 1st derivative dA/dx

solve for x

this will give the value of x that gives the maximum area.

to find y substitute x into the perimeter formula .

Answer 3

\[90=2(x+y)\]\[45=x+y\]\[y=45-x\]\[A=xy=x(45-x)\]\[A(x)=45x-x^2\]\[A'(x)=?\]

Answer 4

From Mathematica using the Constrained Optimization function, Maximize:\[\text{Maximize}[\{w* L,2*w+L==90,w>0,L>0\},\{w,L\}] \]\[\left\{\frac{2025}{2},\left\{w\to \frac{45}{2},L\to 45\right\}\right\} \]2025/2 is the value of the maximum area.