Question

Could someone explain how to manually solve this?

Answer 1

12.85 ft

Explanation:

The total area of the window is the sum of the area of a semicircle with a diameter of x feet, and the area of a square with side lengths of x feet.

The formula for the area of a semicircle with radius r is:

`\boxed{\textsf{Area of a semicircle}=(1)/(2)\pi r^2}`

The formula for the area of a square with side length x is:

`\boxed{\textsf{Area of a square}=x^2}`

As the diameter of the semicircle is equal to the side length of the square, and the diameter of a circle is twice its radius, the radius of the semicircle is x/2 feet.

Therefore, the equation for the area of the window is:

`\textsf{Area of the window}=(1)/(2)\pi \left((x)/(2)\right)^2+x^2`

Given that the total area of the window is 230 ft², we can substitute 230 into the equation and solve for x:

`\begin{aligned}(1)/(2)\pi \left((x)/(2)\right)^2+x^2&=230\\\\(1)/(2)\pi \left((x^2)/(2^2)\right)+x^2&=230\\\\(1)/(2)\pi \left((x^2)/(4)\right)+x^2&=230\\\\(x^2\pi)/(8) +x^2&=230\\\\(x^2\pi)/(8) +(8x^2)/(8)&=230\\\\(x^2\pi+8x^2)/(8)&=230\\\\x^2\pi+8x^2&=1840\\\\x^2(\pi+8)&=1840\\\\x^2&=(1840)/(\pi+8)\\\\x&=\sqrt{(1840)/(\pi+8)}\\\\x&=12.850951169917...\\\\x&=12.85\; \sf ft\;(nearest\;hundredth)\end{aligned}`

As x is equal to the width of the window, the width of the window is 12.85 ft (rounded to the nearest hundredth).