Question

The figure is shown in a rectangle topped with a semicircle. The base of the rectangle is one-third its height. If $h$ is the height of the rectangle, what is the area of the figure?

• A. $\left(\frac{24+\pi }{72}\right)h^2$
• B. $\left(\frac{12+\pi }{72}\right)h^2$
• C. $\left(\frac{\pi }{3}\right)h^2$
• D. $\left(\frac{24+9\pi }{8}\right)h^2$
• E. $\left(\frac{24+9\pi }{4}\right)h^2$

Answer 1

The correct option is A $\left(\frac{24+\pi }{72}\right)h^2$

Given that:

The height of the rectangle is equal to $h.$

The base of the rectangle $=\frac{h}{3}$

The base of the rectangle $=$ Diameter of the semicircle

$\therefore$ Radius of the circle $=\frac{h}{6}$

Now,

Area of the figure $=$ Area of the rectangular part $+$ Area of the semicircular part

$=h\times \frac{h}{3}+\frac{1}{2}\times \pi \times {\left(\frac{h}{6}\right)}^2$

$=\frac{h^2}{3}+\frac{\pi h^2}{72}$

$=\left(\frac{24+\pi }{72}\right)h^2$

Hence, option $A$ is correct.