Question

The radius of the cylinder is half its height and area of the inner part is 616 sq. cm. How many litres of milk approximately can it contain?
A.1.4 L
B.1.5 L
C.1.7 L
D.1.9 L

Answer 1

Hint: Here we are given that, $$r={\displaystyle \frac{h}{2}}$$ and area of inner part of cylinder is given as 616 sq. cm, we use the formula of surface area of cylinder, and substitute the value of r and first find the value of height of cylinder, then using that we find the volume of cylinder.

Complete step-by-step answer:

Here, we consider the cylinder as a right circular cylinder since the base of the given cylinder is a circle.
Let, the height and radius of the cylinder is h and r respectively.
Then, the base area $$=\pi r^2$$ and the whole surface $$=\left(2\pi rh+2\pi r^2\right)$$and the volume$$=\pi r^{2}h$$
Here, given, radius of the cylinder is half its height, i.e. $$r={\displaystyle \frac{h}{2}}$$
We know, the formula that the whole surface of a right cylinder
$$\begin{gathered} =\left(areaoftwobase\right)+\left(areaofthelateralsurface\right) \\ =\left(2\pi rh+2\pi r^2\right) \end{gathered}$$
Here given, the area of the inner part is 616 sq. cm.
$$\begin{gathered} \therefore 2\pi rh+2\pi r^2=616 \\ \Rightarrow 2\pi \left({\displaystyle \frac{h}{2}}\right)h+2\pi {\left({\displaystyle \frac{h}{2}}\right)}^2=616\left[\because r={\displaystyle \frac{h}{2}}\right] \\ \Rightarrow \pi h^2+{\displaystyle \frac{\pi h^2}{4}}=616 \\ \Rightarrow \pi h^2\left(1+{\displaystyle \frac{1}{4}}\right)=616 \\ \Rightarrow \pi h^2\cdot {\displaystyle \frac{5}{4}}=616 \end{gathered}$$
On simplifying and substituting the value of $$\pi$$, and simplify we get,
$$\Rightarrow h^2={\displaystyle \frac{28\times 28}{5}}$$
$$\Rightarrow {\text{h}}^2={\displaystyle \frac{784}{5}}$$
Taking square root on both the sides we get,
$$\Rightarrow h={\displaystyle \frac{28}{\sqrt{5}}}$$
Now, we find the volume of the cylinder,
$$\begin{gathered} =\pi r^{2}h \\ =\pi {\left({\displaystyle \frac{h}{2}}\right)}^{2}h \end{gathered}$$
$$={\displaystyle \frac{\pi }{4}}\cdot h^3$$
On substituting the value of h we get,
$$\begin{gathered} ={\displaystyle \frac{\pi }{4}}\cdot {\left({\displaystyle \frac{28}{\sqrt{5}}}\right)}^3 \\ ={\displaystyle \frac{22}{7}}\times {\displaystyle \frac{28\times 28\times 28}{5\sqrt{5}}}\times {\displaystyle \frac{1}{4}} \\ =1542.7 \end{gathered}$$
$$\therefore$$The volume of the cylinder$$\text{ = 1542}\text{.7 }c{m}^3$$$$={\displaystyle \frac{1542.7}{100}}{m}^3$$$$=1.5m^3=1.5L\left[\because 1L=1m^3\right]$$
So, the cylinder contains 1.5 L milk approximately.
Hence, option (B) is correct.

Note: Here we apply the formula of cylinder, which are, the whole surface area of a right cylinder $$=\left(2\pi rh+2\pi r^2\right)$$units and volume $$=\pi r^{2}h$$units, where r and h are the radius and height of the cylinder.
Note that we have to consider the entire surface area of the cylinder and not just the lateral surface area.