Question

The length of the rectangle is 5cm less than thrice its width. Find the dimensions of the rectangle if its area is 112cm²?

Answer 1

Length: `"16 cm"`
Width: `"7 cm"`

Explanation:

First, start by writing the formula for the area of a rectangle of width `w` and length `l`

`color(blue)(A = l * w)`

Now, you know that if you triple the rectangle's width and subtract 5 cm from the result, you get the rectangle's length.

This means that you can write

`l = 3 * w - 5`

Since you know that the area of the rectangle is equal to `"112 cm"""^3`, you can write a second equation using `l` and `w`

`(3w - 5) * w = 112`

`3w^2 - 5w = 112`

`3w^2 - 5w - 112 = 0`

Use the quadratic formula to find the two solutions to this quadratic equation

`w_(1,2) = ((-5)) +- sqrt((-5)^2 - 4 * 3 * (-112))/(2 * 3)`

`w_(1,2) = (5 +- sqrt(1369))/6`

`w_(1,2) = (5 +- 37)/6`

Since `w` represents the width of the rectangle, the negative solution will have no physical significance. This means that the only valid solution to this quadratic is

`w = (5 + 37)/6 = 42/6 = color(green)("7 cm")`

The length of the rectangle will be

`3 * 7 - 5 = 21 - 5 = color(green)("16 cm")`