Q. 328 - Intermediate Algebra

Question

A rectangular carport has area $150$ square feet. The height of the carport is five feet less than twice its length. Find the height and the length of the carport.

Step-by-Step Solution

Answer

The length and the height of the rectangular carport is $10 feet$ and $15 feet$.

Step 1. Assumption of integers.

Assume $x feet$ to be the length and $2x-5 feet$ to be the height of the carport.

Area of rectangular carport$=length\times height$

Substitute the values in the equation of area,

$$\begin{gathered} 150=x\left(2x-5\right) \\ 150=2x^2-5x \\ 2x^2-5x-150=0 ...... \left(i\right) \end{gathered}$$

Step 2. Factor the equation.

On factoring,

$$\begin{gathered} 2x^2-20x+15x-150=0 \\ 2x\left(x-10\right)+15\left(x-10\right)=0 \\ \left(x-10\right)\left(2x+15\right)=0 \end{gathered}$$

Use the zero product property,

$$\begin{gathered} x-10=0 \\ x=10 \end{gathered}$$

Then,

$$\begin{gathered} 2x+15=0 \\ x=-\frac{15}{2} \end{gathered}$$

As we know, the length of the carport cannot be negative, $x=10$.

Then the height of the carport,

$$\begin{gathered} =2x-5 \\ =2\left(10\right)-5 \\ =15 \end{gathered}$$

Step 3. Check the answers.

Substitute the height and the length in equation (i),

$$\begin{gathered} 150=10\times 15 \\ 150=150 \end{gathered}$$

This is true.

Hence, the length and height is $10 feet$ and $15 feet$.