Question

A rectangular lawn is 24 feet wide by 32 feet long. A sidewalk will be built along the inside edges of all four sides. The remaining lawn will have an area of 425 square feet. How wide will the walk be?

Answer 1

`"width" = "3.5 m"`

Explanation:

Take the width of the side walk as `x`, so the length of the remaining lawn becomes

`l = 32 - 2x`

and the width of the lawn becomes

`w = 24 - 2x`

The area of the lawn is

`A = l * w = (32 - 2x) * (24 - 2x) = 4x^2 -112x + 768`

This is equal to `"425 ft"^2 ->` given

This means that you have

`4x^2 - 112x + 768 = 425`

`4x^2 - 112x + 343 = 0`

This is a quadratic equation and you can solve it using the quadratic formula

`x_(1,2) = (-b +- sqrt(b^2 - 4 * a * c))/(2 * a)" "`, where

`a` is the coefficient of `x^2 ->` `4` in this case
`b` is the coefficient of `x ->` `-112` in this case
`c` is the constant `-> 343` in this case

Out of the two values which you get for `x`, one will be absurd. Discard it and consider the other.

`x_(1,2) = (-(-112) +- sqrt(7056))/(2 * 4)`

`x_(1,2) = (112 +- 84)/8 = { (color(red)(cancel(color(black)(x_1 = 24.5)))), (x_2 = 3.5) :}`

Thewidth of the sidewalk will thus be

`x = "3.5 m"`