Question

A rectangular garden 200 square feet in area is to be fenced o against rabbits. Find thedimensions that will require the least amount of fencing given that one side of the garden isalready protected by a barn

Answer 1

20 ft by 10ft

Explanation:

Area of the garden =200 square feet

Let the dimensions of the garden be x and y

Area, xy=200

[TeX]x=\frac{200}{y}[/TeX]

Since one side is already protected by a barn,

Perimeter of the garden=2x+y

[TeX]P(y)=\frac{2X200}{y}+y[/TeX]

[TeX]P(y)=\frac{400}{y}+y[/TeX]

[TeX]P(y)=\frac{400+y^2}{y}[/TeX]

The minimum point of P(y) is the point where its derivative equals zero.

[TeX]P^{1}(y)=\frac{y^2-400}{y^2}[/TeX]

[TeX]\frac{y^2-400}{y^2}=0[/TeX]

[TeX]y^2-400=0[/TeX]

[TeX]y^2=400[/TeX]

y=20feet

Recall:

xy=200

20x=200

x=10feet

The dimensions of the garden that will require minimum fencing is 20 ft by 10ft