A semicircle is placed on one side of a square so that its d

[LaryssaLacerda]

A semicircle is placed on one side of a square so that its diameter coincides with a side of the square. Find the side length of the square if the total area of the square plus the semicircle is 200 square inches.

 

[soroban]

Re: Math problem!!! Geometry

Hello, Laryssa!

Did you make a sketch?

A semicircle is placed on one side of a square so that its diameter
coincides with a side of the square.
Find the side of the square if the total area of the square and semicircle is 200 in².

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                x

Let \(\displaystyle x\) = side of the square.
The area of the square is: \(\displaystyle x^2\)

The radius of the semicircle is: \(\displaystyle \frac{x}{2}\)
The area of the semicircle is: \(\displaystyle \,\frac{1}{2}\cdot\pi\left(\frac{x}{2}\right)^2\:=\:\frac{\pi}{8}x^2\)

Then the total area is: \(\displaystyle x^2\,+\,\frac{\pi}{8}x^2\:=\:200\)


Solve for \(\displaystyle x.\)

Multiply by 8: \(\displaystyle \:8x^2 + \pi x^2\:=\:1600\;\;\Rightarrow\;\;(\pi\,+\,8)x^2\:=\:1600 \;\;\Rightarrow\;\;x^2\:=\:\frac{1600}{\pi\,+\,8}\)

Therefore: \(\displaystyle \:x\:=\:\L\sqrt{\frac{1600}{\pi\,+\,8}}\)\(\displaystyle \:\approx\;12\) inches.

 

[LaryssaLacerda]

thanks

Thank-you so much.. you really helped me alot!!!