Question

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

The lateral surface area $S_1$ of a right circular cylinder with an altitude of $h_1=40\ cm$ and base diameter of $d=30\ cm$ is equal to $S_1=\pi \cdot d\cdot h=\pi\cdot 40\cdot 30=1200\pi$ ($cm^2$ )

The lateral surface area $S_2$ of a right circular cone with an altitude of $h_2=60\ cm$ and base diameter of $d=30\ cm$ is equal to $S_2=\pi \cdot r\cdot l$, where $r=\frac{d}{2}=15\ (cm)$ is the radius of the base and $l=\sqrt{(h_2)^2+r^2}=\sqrt{60^2+15^2}=\sqrt{3600+225}=\sqrt{3825}=15\sqrt{17}\ (cm)$ is the slant height of the cone.  Therefore, $S_2=\pi\cdot 15\cdot 15\sqrt{17}=225\sqrt{17}\pi\ (cm^2)$.

The lateral surface area of a right circular cylinder with a right circular cone at its top is$1200\pi+225\sqrt{17}\pi=75\pi(16+3\sqrt{17})\ (cm^2).$