Question

The volume and area of the curved surface of the right circular cylinder is $1540m^3$ and $440m^2$ respectively. Then find the radius of base and height of the cylinder.

Answer 1

Hint: Now we know that the curved surface area of right circular cylinder is given by $\pi r^{2}h$ and the curved surface of right circular cylinder is given by $2\pi rh$ . Hence using this we will get two equations in r and h. we will solve them simultaneously to find r and h.

Complete step by step answer:
Now consider a right circular cylinder

We know that the Volume of curved right circular cylinder is given by formula $\pi r^{2}h$ where r is the radius of the cylinder and h is the height of the cylinder.
Now the volume of cylinder is given as $1540m^3$
Hence we have $\pi r^{2}h=1540m^3$
Let this be called equation (1)
$\pi r^{2}h=1540m^3.........................\left(1\right)$
Now we also have formula for curved surface area of cylinder which is $2\pi rh$
Also the curved surface area is given to be $440m^2$
Hence we have $2\pi rh=440$
Let this be called equation (2)
$2\pi rh=440m^2..............\left(2\right)$
Now let us divide equation (1) by equation (2). Hence we will get
${\displaystyle \frac{\pi r^{2}h}{2\pi rh}}={\displaystyle \frac{1540}{440}}$
Now cancelling the common terms we get ${\displaystyle \frac{r}{2}}={\displaystyle \frac{1540}{440}}$
Now if we divide 1540 and 440 by 220 and hence we get
${\displaystyle \frac{r}{2}}={\displaystyle \frac{7}{2}}$
Now multiplying the whole equation by 2 we get r = 7
Hence r = 7m
Now substituting r = 7 in equation (2) we get
$2\pi \left(7\right)h=440$
Now for simplicity let us take $\pi ={\displaystyle \frac{22}{7}}$
Hence we have
$\begin{array}{rl}& 2\times {\displaystyle \frac{22}{7}}\times 7\times h=440\\ & \Rightarrow 2\times 22\times h=440\\ & \Rightarrow 44\times h=440\\ & \Rightarrow h=10\end{array}$
Hence we get h = 10m

Hence the value of r = 7m and h = 10m.

Note: Note that the curved surface area and total surface area are different for the cylinder. Curved surface area of the cylinder is given by $2\pi rh$ and total surface area is given by $2\pi rh+2\pi r^2$ . Also note that $\pi$ is an irrational number and cannot be expressed in the form of ${\displaystyle \frac{p}{q}}$ for simplicity we take it as ${\displaystyle \frac{22}{7}}$.