Question

The perimeter of a rectangular goat pen is 28 meters. The area is 45 square meters. Whatare the dimensions of the pen?

Answer 1

Given the question in the image, the following are the solution steps to answer the question.

STEP 1: Write the given information

`\begin{gathered} Perimeter=2l+2b=28m \\ Area=lb=45m^2 \\ \\ where\text{ l is the length, b is the breadth} \end{gathered}`

STEP 2: Label the two equations

`\begin{gathered} 2l+2b=28------equation\text{ 1} \\ lb=45----equation\text{ 2} \end{gathered}`

STEP 3: Solve for the missing values

Isolate l in equation 1

`\begin{gathered} 2(l+b)=28 \\ l+b=(28)/(2)=14 \\ l=14-b-----equation\text{ 3} \end{gathered}`

Substitute 14-b for l in equation 2

`\begin{gathered} (14-b)\cdot b=45 \\ 14b-b^2=45 \\ We\text{ have the quadratic equation} \\ -b^2+14b-45=0 \end{gathered}`

Solve the equation quadratically

`\begin{gathered} -b^(2)+14b-45=0 \\ -b^2+9b+5b-45=0 \\ -b(b-9)+5(b-9)=0 \\ (-b+5)(b-9)=0 \\ -b+5=0,b=5 \\ b-9=0,b=9 \\ \\ b=5,b=9 \end{gathered}`

Substitute the values into equation 3,

`\begin{gathered} l=14-b \\ when\text{ b = 5} \\ l=14-5=9 \\ When\text{ b = 9} \\ l=14-9=5 \end{gathered}`

Hence, the dimensions of the pen is given as:

`9m\text{ }by\text{ }5m`