Question

It costs Rs. $\mathbf{2200}$ to paint the inner curved surface of a cylindrical vessel $\mathbf{10}\mathbf{}\mathbf{m}$ deep. If the cost of painting is at the rate of Rs. $\mathbf{20}$ per ${\mathbf{m}}^{\mathbf{2}}$, find

(i) inner curved surface area of the vessel,

(ii) radius of the base,

(iii) capacity of the vessel,

Answer 1

Step 1: Given information.

Height of cylinder, $h=10\mathrm{m}$

Cost of painting rate$=\mathrm{Rs}.20\mathrm{per}{\mathrm{m}}^2$

Step 2: Find the inner curved surface area of the vessel.

Cost of painting for $1{\mathrm{m}}^2=\mathrm{Rs}.20$

Cost to paint the inner curved surface area of the cylinder$=\mathrm{Rs}.2200$

So, the lateral surface area:

$$\begin{gathered} \mathrm{L}.\mathrm{S}.\mathrm{A}.=\frac{2200}{20} \\ =110{\mathrm{m}}^2 \end{gathered}$$

Step 3: Find the radius of the base.

Let $r$ be the radius of the cylinder.

The lateral surface area of the cylinder$=2\mathrm{\pi }rh$

$$\begin{gathered} \Rightarrow 110=2\mathrm{\pi }rh\mathit{} \\ \Rightarrow 110=2\times \frac{22}{7}\times r\times 10 \\ \Rightarrow r=\frac{110\times 7}{2\times 22\times 10} \\ \therefore r=1.75\mathrm{m} \end{gathered}$$

Step 3: Find the capacity of the vessel:

The volume of the cylinder,

$$\begin{gathered} V=\mathrm{\pi }{r}^{\mathit{2}}h \\ =\frac{22}{7}\times {\left(1.75\right)}^2\times 10 \\ =\frac{22}{7}\times 3.0625\times 10 \\ =96.25{\mathrm{m}}^3 \\ =96.25\mathrm{kl}\left[1\mathrm{kl}=1{\mathrm{m}}^3\right] \end{gathered}$$

Therefore, the inner curved surface area of the vessel is$110{\mathrm{m}}^2$, the radius of the base is $1.75\mathrm{m}$, and the capacity of the vessel is $96.25\mathrm{kl}$.