Q7. - Algebra 1

Question

Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper.

Mickey has $180$ feet of fencing that she wants to use to enclose a play area for her puppy. She will use her house as one of the sides of the region.

1. If she makes the play area $x$ feet deep as shown in the figure, write a polynomial in standard form to represent the area of the region.
2. How many square feet of area will the puppy have to play in if Mickey makes it $40$ feet deep?

Step-by-Step Solution

Answer

1. If she makes the play area $x$ feet deep as shown in the figure then a polynomial in standard form to represent the area of the region is $-2x^2+180x$.
2. The square feet of area that will the puppy have to play in if Mickey makes it 40 feet deep is $4000$.

Step 1. Observe the given diagram.

The given diagram is:

From the given diagram it can be noticed that the length of the rectangle is $\left(180-2x\right)\text{ft}$ and the breadth of the rectangle is $x\text{ ft}$.

Step 2. Write the polynomial in standard form to represent the area of the region.

The area $\left(A\right)$ of the rectangle is given by:

$A=\text{length}\times \text{breadth}$

Step3. Calculation

1. Therefore, a polynomial in standard form to represent the area of the given region is given by:

$$\begin{gathered} A=\text{length}\times \text{breadth} \\ =\left(180-2x\right)\text{ft}\times x\text{ ft} \\ =\left(180\times x-2x\times x\right){\text{ft}}^2 \\ =\left(180x-2x^2\right){\text{ft}}^2 \\ =\left(-2x^2+180x\right){\text{ft}}^2 \end{gathered}$$

Therefore, if she makes the play area $x$ feet deep as shown in the figure then a polynomial in standard form to represent the area of the region is $-2x^2+180x$.

     b A polynomial in standard form to represent the area of the given region is $-2x^2+180x$.

 

It is given that the depth is $40$ feet. That implies the breadth of the rectangle is $40$ feet. Therefore, the value of $x$ is $40$.

Substitute $40$ for $x$ in $A=-2x^2+180x$.

Therefore, it is obtained that:

$$\begin{gathered} A=-2x^2+180x \\ =-2{\left(40\right)}^2+180\left(40\right) \\ =-2\left(1600\right)+7200 \\ =-3200+7200 \\ =4000 \end{gathered}$$

Therefore, the square feet of area that will the puppy have to play in if Mickey makes it $40$ feet deep is $4000$.