Question

solve. you may use a calculator.
the swimming pool at lake spring high school is a square with a width of 65 meters. around the perimeter of the pool is a tiled floor that extends ( x ) meters from the pool on the left and right side and ( x^2 + 2x ) meters on the other two sides.

1. find the combined area of the pool and tiled floor.
2. find the area of the tiled floor.

Explanation:

1. Find the combined area of the pool and tiled floor.

Step1: Determine the length and width of the combined area

The pool is a square with width 65 meters. The tiled floor extends \( x \) meters on the left and right, so the total width of the combined area (pool + tiled floor) is \( 65 + 2x \) meters (since it extends \( x \) on both left and right). The tiled floor extends \( x^{2}+2x \) meters on the other two sides (let's say top and bottom), so the total length of the combined area is \( 65 + 2(x^{2}+2x) \) meters. Simplify the length: \( 65 + 2x^{2}+4x=2x^{2}+4x + 65 \) meters.

Step2: Calculate the combined area

The combined area is the product of the length and the width of the combined region (since it's a rectangle, as the pool is square and the extensions are uniform in their respective directions). So the area \( A \) is:
\[
A=(2x^{2}+4x + 65)(65 + 2x)
\]
We can expand this using the distributive property (FOIL method for polynomials):
\[

$$\begin{align*} A&=(2x^{2})(65)+(2x^{2})(2x)+(4x)(65)+(4x)(2x)+(65)(65)+(65)(2x)\\ &=130x^{2}+4x^{3}+260x + 8x^{2}+4225+130x\\ &=4x^{3}+(130x^{2}+8x^{2})+(260x + 130x)+4225\\ &=4x^{3}+138x^{2}+390x + 4225 \end{align*}$$

\]

Step1: Find the area of the pool

The pool is a square with side length 65 meters, so its area \( A_{pool} \) is \( 65\times65 = 4225 \) square meters.

Step2: Subtract the pool's area from the combined area

We already found the combined area (pool + tiled floor) in part 1 as \( 4x^{3}+138x^{2}+390x + 4225 \). The area of the tiled floor \( A_{tiled} \) is the combined area minus the area of the pool:
\[
A_{tiled}=(4x^{3}+138x^{2}+390x + 4225)-4225
\]
Simplify this expression:
\[
A_{tiled}=4x^{3}+138x^{2}+390x
\]

Answer:

The combined area of the pool and tiled floor is \( \boldsymbol{4x^{3}+138x^{2}+390x + 4225} \) square meters.

2. Find the area of the tiled floor.