Question

You are planning a rectangular patio with length that is `7` `ft` less than three times its width. The area of the patio is `120` `ft^2`. What are the dimensions of the patio?

Answer 1

The dimensions of the rectangular patio are width = `7.67` and the length = `16.01` ft.

Explanation:

Since the length is defined by the width, let `x` represent the width. This means that the length will be represented by the expression `3x - 7`. The area of a rectangle is found by multiplying its length by its width. Substitute and solve for `w`.

`A = lw`
`120 = (3x - 7)x`
`120 = 3x^2 - 7x`
`120 - 120 = 3x^2 - 7x - 120`
`0 = 3x^2 - 7x - 120`

Now that the equation is simplified and in standard form, use the Quadratic Formula to find the possible solutions for `x`. The Quadratic Formula is `x = (-b +- sqrt(b^2 - 4ac))/(2a)`. For this situation, `a = 3`, `b = -7`, and `c = -120`.

`x = (-(-7) +- sqrt((-7)^2 - 4(3)(-120)))/(2*3)`
`x = (7 +- sqrt(49 +1440))/6`
`x = (7 +- sqrt(1489))/6`
`x ~~ (7+-39)/6`
`x ~~ (7 + 39)/6` or `x ~~ (7 - 39)/6`
`x ~~ 46/6` or `x ~~ -32/6`
`x ~~ 7.67` or `x ~~ -5.33`

Since distances (width, in this case) are not negative, disregard `-5.33`. The width of the rectangular patio is approximately `7.67`. Use this value to find the length of the patio.

`l = 3(7.67) - 7`
`l = 23.01 - 7`
`l = 16.01`

The dimensions of the rectangular patio are width = `7.67` ft and length = `16.01` ft.