Question

A decorative window is made up of a rectangle with semicircles at either end. The ratio of AD to AB is 3:2 and AB is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircles?

PLS HELP ME QUICK, I HAVE TO BE DONE BY 3!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Answer 1

Answer: The required ratio will be

`84:1034`

Explanation:

Since we have given that

Ratio of AD to AB is 3:2

Length of AB = 30 inches

So, it becomes

`2x=30\\\\x=(30)/(2)=15\ inches`

So, Length of AD becomes

`3x=3* 15=45\ inches`

Now, at either end , there is a semicircle.

Radius of semicircle along AB is given by

`(30)/(2)=15\ inches`

So, Area of semicircle along AB and CD is given by

`2* (\pi r^2)/(2)\\\\=(22)/(7)* 15* 15\\\\=(4950)/(7)\ in^2`

Radius of semicircle along AD is given by

`(45)/(2)=22.5\ inches`

Area of semicircle along AD and BC is given by

`2* (1)/(2)\pi r^2\\\\=(22)/(7)* (45)/(2)* (45)/(2)\\\\=(445500)/(28)\ in^2`

And the combined area of the semicircles is given by

`(4950)/(7)+(445500)/(28)\\\\=(465300)/(28)\ in^2`

Area of rectangle is given by

`Length* width\\\\=AD* AB\\\\=45* 30\\\\=1350\ in^2`

Hence, Ratio of the area of the rectangle to the combined area of the semicircles is given by

`1350:(465300)/(28)\\\\=1350* 28:465300\\\\=37800:465300\\\\=84:1034`

Hence, the required ratio will be

`84:1034`