Question

The surface area of a cylinder varies jointly as the radius, and the sum of the radius and the height. A cylinder with height $8cm$ and radius $4cm$ has a surface area of $96\pi c{m}^2$. Find the surface area of a cylinder with radius $3cm$ and height $10cm$

Answer 1

It is given that the area $A$ of a cylinder varies jointly as the radius $r$ and the sum of the radius and height $(r+h)$ and$A=96\pi$ cm${}^2$ when $r=4$ cm and $h=8$ cm, therefore,

$A=kr(r+h)\Rightarrow 96\pi =k\times 4(4+8)\Rightarrow 96\pi =k\times 4\times 12\Rightarrow 48k=96\pi \Rightarrow k=\frac{96\pi }{48}=2\pi$

Thus, the general equation is $A=2\pi r(r+h)$.

Since $r=3$ cm and $h=10$ cm, therefore,

$A=2\pi r(r+h)=2\pi \times 3(3+10)=2\pi \times 3\times 13=78\pi$

Hence, the surface area of the cylinder with radius $3$ cm and height $10$ cm is $78\pi$ cm${}^2$.