Question

A rectangular lot has an area of 240m square .what is the width of the lot if it requires 64m of fencing materials to enclose it?

Answer 1

x, y - the lengths of the sides of the lot

The area of a rectangle is the product of the lengths of its sides. The area is 240 m².

`xy=240`

It requires 64 m of fencing materials to enclose it, so its perimeter is 64 m. The perimeter of a rectangle is two times the sum of the lengths of its sides.

`2(x+y)=64 \ \ \ \ \ \ |/ 2 \\ x+y=32 \ \ \ \ \ \ \ \ \ \ |-x \\ y=32-x`

Substitute 32-x for y in the first equation:

`x(32-x)=240 \\ 32x-x^2=240 \\ -x^2+32x-240=0 \\ -x^2+12x+20x-240=0 \\ -x(x-12)+20(x-12)=0 \\ (-x+20)(x-12)=0 \\ -x+20=0 \ \lor \ x-12=0 \\ x=20 \ \lor \ x=12`


`y=32-x \\ \hbox{for } x=20: \\ y=32-20=12 \\ \\ \hbox{for } x=12: \\ y=32-12=20`

The dimensions of the lot are 12 m by 20 m.
The width of the lot is 12 m.