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.

Answer 1

We are given that the area of a rectangle is 150 square feet. If "h" is the height and "l" is the length then the area is given by:

$hl=150$

We are also given that the height is 5 feet less than twice its length, this can be written mathematically as:

$h=2l-5$

Now we can replace the value of "h" in the equation for the area:

$(2l-5)l=150$

Now we use the distributive property:

$2l^2-5l=150$

Now we have a quadratic equation that can be written as:

$2l^2-5l-150=0$

We can factor this equation to determine the values of "l". We multiply and divide by 2:

$\frac{4l^2-5(2l)-300}{2}=0$

Factoring in the numerator:

$\frac{(2l-20)(2l+15)}{2}=0$

Now we take common factor in the first parenthesis in the numerator:

$\frac{2(l-10)(2l+15)}{2}=0$

Simplifying:

$(l-10)(2l+15)=0$

Now we set each factor to zero:

$\begin{gathered} l_1-10=0 \\ l_1=10 \\ 2l_2+15=0 \\ l_2=-\frac{15}{2} \end{gathered}$

We take the positive value, therefore, the length of the rectangle is 10 feet. Now we replace this value in the equation for the height.

$h=2(10)-5$

Solving we get:

$\begin{gathered} h=20-5 \\ h=15 \end{gathered}$

Therefore, the height of the rectangle is 15 feet.