Answer: The required ratio will be
[tex]84:1034[/tex]
Step-by-step explanation:
Since we have given that
Ratio of AD to AB is 3:2
Length of AB = 30 inches
So, it becomes
[tex]2x=30\\\\x=\frac{30}{2}=15\ inches[/tex]
So, Length of AD becomes
[tex]3x=3\times 15=45\ inches[/tex]
Now, at either end , there is a semicircle.
Radius of semicircle along AB is given by
[tex]\frac{30}{2}=15\ inches[/tex]
So, Area of semicircle along AB and CD is given by
[tex]2\times \frac{\pi r^2}{2}\\\\=\frac{22}{7}\times 15\times 15\\\\=\frac{4950}{7}\ in^2[/tex]
Radius of semicircle along AD is given by
[tex]\frac{45}{2}=22.5\ inches[/tex]
Area of semicircle along AD and BC is given by
[tex]2\times \frac{1}{2}\pi r^2\\\\=\frac{22}{7}\times \frac{45}{2}\times \frac{45}{2}\\\\=\frac{445500}{28}\ in^2[/tex]
And the combined area of the semicircles is given by
[tex]\frac{4950}{7}+\frac{445500}{28}\\\\=\frac{465300}{28}\ in^2[/tex]
Area of rectangle is given by
[tex]Length\times width\\\\=AD\times AB\\\\=45\times 30\\\\=1350\ in^2[/tex]
Hence, Ratio of the area of the rectangle to the combined area of the semicircles is given by
[tex]1350:\frac{465300}{28}\\\\=1350\times 28:465300\\\\=37800:465300\\\\=84:1034[/tex]
Hence, the required ratio will be
[tex]84:1034[/tex]
