Question

a family needs to build fencing around their rectangular home and square swimming pool, depicted below. pool (2x) yd home (2 + 5x) yd (3 + 10x) yd the total amount of fencing they need can be written as \boxed{}x + 10. if x=5; the family needs \boxed{} yards of fencing for their home. the family needs \boxed{} yards of fencing for their pool.

Explanation:

First, find the coefficient for the total fencing expression (assuming it's the sum of home and pool perimeters)

Step 1: Calculate the perimeter of the home (rectangle)

The perimeter of a rectangle is \( P = 2\times(\text{length} + \text{width}) \). For the home, length is \( (3 + 10x) \) yd and width is \( (2 + 5x) \) yd. So:
\[

$$\begin{align*} P_{\text{home}}&= 2\times[(3 + 10x)+(2 + 5x)]\\ &= 2\times(5 + 15x)\\ &= 10 + 30x \end{align*}$$

\]

Step 2: Calculate the perimeter of the pool (square)

The perimeter of a square is \( P = 4\times\text{side} \). For the pool, side is \( (2x) \) yd. So:
\[
P_{\text{pool}} = 4\times(2x)= 8x
\]

Step 3: Calculate the total fencing (sum of home and pool perimeters)

\[

$$\begin{align*} P_{\text{total}}&= P_{\text{home}}+ P_{\text{pool}}\\ &= (10 + 30x)+ 8x\\ &= 10 + 38x \end{align*}$$

\]
So the coefficient of \( x \) in the total fencing expression is \( 38 \).

Now, when \( x = 5 \):

Step 4: Fencing for the home

Substitute \( x = 5 \) into \( P_{\text{home}} = 10 + 30x \):
\[
P_{\text{home}}= 10 + 30\times5 = 10 + 150 = 160
\]

Step 5: Fencing for the pool

Substitute \( x = 5 \) into \( P_{\text{pool}} = 8x \):
\[
P_{\text{pool}}= 8\times5 = 40
\]

Answer:

s:

• The total amount of fencing coefficient: \( \boldsymbol{38} \)
• Fencing for home (when \( x = 5 \)): \( \boldsymbol{160} \)
• Fencing for pool (when \( x = 5 \)): \( \boldsymbol{40} \)