Question

A wax Candle is in the shape of right circular cone with base radius 5 cm and height 12cm. It takes 1 hour 40 minutes to burn completely. After 25/2 minutes of burning the candle is reduced to a frustum with the height of h cm. Find the volumof the candle before burning, the total surface area of the candle before burning and the value of h.​

Answer 1

Volume = 314.28 cm³, Surface area = 282.85 cm² and h = 6 cm.

Step-by-step explanation:

The original shape of the candle is a right circular cone with a base radius  of 5 cm and a height of 12 cm.

So, the volume of the candle before burning was $\frac{1}{3}\pi r^{2} h = \frac{1}{3} \times (\frac{22}{7}) \times 5^{2} \times 12 = 314.28$ cubic cm.

And the total surface area of the candle before burning was $\pi r\sqrt{r^{2} + h^{2}} + \pi r^{2} = (\frac{22}{7})\times 5 \times \sqrt{5^{2} + 12^{2}} + (\frac{22}{7})5^{2} = 282.85$ cm².

Now, in 1 hr and 40 minutes i.e. $\frac{5}{3}$ hours the 314.28 cm³ volume burns.

So, in $\frac{25}{2}$ minutes i.e. $\frac{5}{24}$ hours $\frac{314.28 \times 3 \times 5}{5 \times 24} = 39.285$ cm³ volume burns.

Now, this small volume will be a small cone at the top of the frustum.

Let the radius of the small cone is r and the height is x.

Then $\frac{x}{r} = \frac{H}{R} = \frac{12}{5}$

⇒ $x = \frac{12r}{5}$ ....... (1)

Now, volume = $\frac{1}{3} \pi r^{2} x = 39.285$

⇒ $r^{2}x = 37.5$ ........ (2)

Now, solving equations (1) and (2) we get,

$r^{3} = 15.63$

r = 2.5 cm

So, x = 6 cm {From equation (1)}

So, h = H - x = 12 - 6 = 6 cm (Answer)