Answer to Question #148932 in Geometry for solid mensuration

Question

8. A right circular cylinder with an altitude of 40 cm and base diameter of 30cm has a right circular cone at its top with the same base. If the cone is 60cm high, determine its lateral area in cm2.

Answer 1

Solution. The lateral area of the body is equal to the sum of the lateral area of the right circular cylinder and the lateral area of the right circular cone

$$S=S_{cylinder}+S_{cone}$$

The lateral area of the right circular cylinder

$$S_{cylinder}=\pi\times dH_1$$

where d=30cm is base diameter; H₁=40cm is altitude of the right circular cylinder.

$$S_{cylinder}=\pi\times 30\times 40=1200\pi cm^2$$

The lateral area of the right circular cone

$$S_{cone}=\pi\times rL$$

where r=d/2=15cm is cone base radius, L is slant height of the cone.

$$L=\sqrt{r^2+h^2}=\sqrt{15^2+60^2}=\sqrt{3825}=5\sqrt{153}$$

where h=60cm is high oh the right circular cone.

$$S_{cone}=\pi\times 15\times 5\sqrt{153}=75\pi\sqrt{153} cm^2$$

As result

$$S=1200\pi+75\pi\sqrt{153} \approx 6684 cm^2$$

Answer.

$$S=1200\pi+75\pi\sqrt{153} \approx 6684 cm^2$$