Question

Find the volume and surface area of the composite figure. Give your answer in terms of pi

Answer 1

Given:

A diagram of a composite figure.

Radius of cone and hemisphere is 8 cm.

Height of the cone is 15 cm.

To find:

The volume and the surface area of the composite figure.

Solution:

Volume of a cone is:

$V_1=\dfrac{1}{3}\pi r^2h$

Where, r is the radius and h is the height of the cone.

Putting $r=8,h=15$ in the above formula, we get

$V_1=\dfrac{1}{3}\pi (8)^2(15)$

$V_1=\pi (64)(5)$

$V_1=320\pi$

Volume of the hemisphere is:

$V_2=\dfrac{2}{3}\pi r^3$

Where, r is the radius.

Putting $r=8$, we get

$V_2=\dfrac{2}{3}\pi (8)^3$

$V_2=\dfrac{1024}{3}\pi$

$V_2\approx 341.3\pi$

Now, the volume of the composite figure is:

$V=V_1+V_2$

$V=320\pi +341.3\pi$

$V=661.3\pi$

The volume of the composite figure is 661.3π cm³.

The curved surface area of a cone is:

$A_1=\pi r\sqrt{h^2+r^2}$

Where, r is the radius and h is the height of the cone.

Putting $r=8,h=15$ in the above formula, we get

$A_1=\pi (8)\sqrt{(15)^2+(8)^2}$

$A_1=\pi (8)\sqrt{289}$

$A_1=\pi (8)(17)$

$A_1=136 \pi$

The curved surface area of the hemisphere is:

$A_2=2\pi r^2$

Where, r is the radius.

Putting $r=8$, we get

$A_2=2\pi (8)^2$

$A_2=2\pi (64)$

$A_2=128\pi$

Total surface area of the composite figure is:

$A=A_1+A_2$

$A=136\pi +128\pi$

$A=264\pi$

The total surface area of the composite figure is 264π cm².

Therefore, the correct option is A.