Question

Milk World Ltd is a government-owned milk distribution centre. Earlier they used to distribute milk in bottles as shown in the figure for Rs 8 per bottle. Now they have decided to reduce the price per bottle by 50 paise by reducing the size of the bottle. However, they do not want their old customers to notice this change in the size of the bottle so they plan to reduce the circumference of the opening of the bottle without changing the height of the bottle. What is the new circumference of the opening?

• A. $6\pi$ cm
• B. $8\pi$ cm
• C. $3.5\pi$ cm
• D. None of these

Answer 1

The correct option is D None of these

Volume of bottle $=\pi R^{2}H+\frac{1}{3}\pi h(R^2+r^2+R.r)$

$=\pi \times 6\times 6\times 4+\frac{1}{3}\pi \times 3(6^2+4^2+6\times 4)$

$=144\pi +76\pi =220\pi {cm}^3$

% reduction in price $=\frac{0.50}{8}\times 100=\frac{25}{4}$%

$\therefore$ Reduction in frustum volume $=76\pi -\frac{25}{4}$% of $220\pi$

$=76\pi -\frac{25}{400}\times 220\pi$

$=76\pi -\frac{55\pi }{4}$

$=\frac{249\pi }{4}$

$\therefore \frac{1}{3}\pi \times 3(6^2+r^2+6r)=\frac{249\pi }{4}$

$\Rightarrow 36+r^2+6r=\frac{249}{4}\Rightarrow 144+4r^2+24r=249$

$\Rightarrow {4r}^2+24r-105=0$

The value of $r$ is $2.94cm$.