Question

The cost of painting the total surface area of a cone at $25$ paise per cm${}^2$ is Rs. $176$. Find the volume of the cone, if its slant height is $25$ cm.

• A. $640c{m}^3$
• B. $1232c{m}^3$
• C. $1543c{m}^3$
• D. $2064c{m}^3$

Answer 1

The correct option is B $1232c{m}^3$

Given total cost $=Rs.176$

Rate of charge = 25 paise per sq.cm

The total cost of painting cone = Total surface area of cone $\times$ Rate of charge

$\Rightarrow$ Total surface area of cone = $\frac{Totalcostofpaintingcone}{Rateofcharge}$

= $\frac{176}{0.25}=704c{m}^2$

Let the radius of cone be $rcm$

slant height $\left(l\right)=25cm$

Total surface area of cone =$\pi r(l+r)=704$

$\Rightarrow \frac{22}{7}\times r(r+25)=704$

$\Rightarrow r^2+25r=\frac{704\times 7}{22}$

$\Rightarrow r^2+25r=224$

$\Rightarrow r^2+32r-7r-224=0$

$\Rightarrow r(r+32)-7(r+32)=0$

$\Rightarrow (r+32)(r-7)=0$

Then $r=7$ and $r=-32$

Discarding negative value as radius cannot be negative

Therefore, radius of cone$=7cm$

By Pythagoras Theorem,

height of cone = $\sqrt{l^2-r^2}$

=$\sqrt{(25{)}^2-(7{)}^2}$

=$\sqrt{625-79}$

=$\sqrt{576}=24$ cm

Therefore, volume of cone = $\frac{1}{3}\pi r^{2}h$

=$\frac{1}{3}\times \frac{22}{7}\times (7{)}^2\times 24$

=$22\times 7\times 8$

=$1232c{m}^3$