Question

A rectangle's base is 2 in shorter than five times its height. The rectangle's area is 115 in². Find this rectangle's dimensions.

The rectangle's height is
The rectangle's base is

Answer 1

Height = 5 in

Base = 23 in

Explanation:

`\boxed{\textsf{Area of a rectangle} = \sf Base * Height}`

Let x be the height of the rectangle.

Given values:

• Height = x in
• Base = (5x - 2) in
• Area = 115 in²

Substitute the values into the formula for area and solve for x:

`\begin{aligned}\sf Area & = \sf Base * Height\\\\115&=x(5x-2)\\115&=5x^2-2x\\5x^2-2x-115&=0\\5x^2-25x+23x-115&=0\\5x(x-5)+23(x-5)&=0\\(5x+23)(x-5)&=0\\\\\implies 5x+23&=0 \implies x=-(23)/(5)\\\implies x-5&=0 \implies x=5\end{aligned}`

As length is positive, x = 5.

To find the rectangle's dimensions, substitute the found value of x into the expressions for the height and base:

`\implies \sf Height=5\;in`

`\begin{aligned}\implies \sf Base&=\sf 5(5)-2\\&=\sf 25-2\\&=\sf 23\;in\end{aligned}`

Answer 2

• The rectangle's height is 5 in,
• The rectangle's base is 23 in.

Explanation:

Let the dimensions are b and h.

The area of rectangle is the product of two dimensions.

We have:

• b = 5h - 2,
• bh = 115 in².

Solve the equation by substitution:

• h(5h - 2) = 115
• 5h² - 2h = 115
• 5h² - 2h - 115 = 0
• 5h² - 25h + 23h - 115 = 0
• 5h(h - 5) + 23(h - 5) = 0
• (h - 5)(5h + 23) = 0

• h - 5 = 0 and 5h + 23 = 0
• h = 5 and h = - 23/5

The second root is discarded as negative.

• The height is 5 in,
• The base is: 5*5 - 2 = 23 in