Question

the total area of two square windows is 1,025 in.². each side of the larger window is 5 in. longer than the sides of the smaller window. how long are the sides of the smaller window?
$a = s^2$
○ 13.5 in.
○ 20 in.
○ 25 in.
○ 31.6 in.

Explanation:

Step1: Define variable for smaller window side

Let the side length of the smaller square window be \( s \) inches. Then the side length of the larger square window is \( s + 5 \) inches.

Step2: Write area equation for total area

The area of a square is \( A = s^2 \). So the area of the smaller window is \( s^2 \) and the area of the larger window is \( (s + 5)^2 \). The total area of the two windows is 1025 \( \text{in}^2 \), so we have the equation:

$$ s^2+(s + 5)^2=1025 $$

Step3: Expand and simplify the equation

Expand \( (s + 5)^2 \): \( (s + 5)^2=s^2+10s + 25 \)
Substitute into the equation:

$$ s^2+s^2+10s + 25=1025 $$

Combine like terms:

$$ 2s^2+10s+25 - 1025=0 $$

$$ 2s^2+10s - 1000=0 $$

Divide the entire equation by 2:

$$ s^2+5s - 500=0 $$

Step4: Solve the quadratic equation

We can solve the quadratic equation \( s^2+5s - 500=0 \) using the quadratic formula \( s=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 5 \), and \( c=- 500 \).
First, calculate the discriminant \( D=b^2 - 4ac=(5)^2-4\times1\times(-500)=25 + 2000 = 2025 \)
Then, \( s=\frac{-5\pm\sqrt{2025}}{2\times1}=\frac{-5\pm45}{2} \)
We have two solutions:
\( s=\frac{-5 + 45}{2}=\frac{40}{2}=20 \) and \( s=\frac{-5-45}{2}=\frac{-50}{2}=-25 \)
Since the side length cannot be negative, we take \( s = 20 \)

Answer:

20 in.