Question

The ratio of diameter and height of a right circular cylinder is 4 : 3. If diameter of the cylinder get reduced by 25% then its total surface area reduced to 318.5 square meter. What is the circumference of the base of the cylinder. (a) 28 pi m (b) 14 pi m (c) 35 pi m (d) 7 pi m (e) none of these

Answer 1

Let the diameter be 4y and height be 3y

Since the diameter = 4y

=> Radius = 4y/2 = 2y



So the total surface area = 2πr(h + r)

=> Total surface area = 2π2y(3y + 2y)

=> Total surface area = 4yπ(5y)

=> Total surface area = 20y²π

Now the diameter is decreased by 25%

=> New diameter = 4y - 25% of 4y

=>
$4y - \frac{25}{100} \times 4y \\ \\ = > 4y - \frac{1}{4} \times 4y \\ \\ = > 4y - y \\ \\ = > 3y$


New diameter = 3y

=> Radius = 3y/2

=> Total surface area =
$2\pi \frac{3y}{2} ( \frac{3y}{2} + 3y)$
$= > 3y\pi( \frac{3y + 6y}{2} )$
=>
$3y\pi( \frac{9y}{2} )$


$= > \frac{27 {y}^{2} }{2} \pi$


Given that new total surface area is 318.5 less than the old surface area

=>
$20 {y}^{2} \pi - \frac{27 {y}^{2} }{2} \pi = 318.5 \\ \\ = > \frac{40y {}^{2} }{2} \pi - \frac{27 {y}^{2} }{2} \pi = 318.5 \\ \\ = > \frac{13y {}^{2} }{2} \pi = 318.5 \\ \\ = > {y}^{2} = 318.5 \times \frac{2}{13\pi} \\ \\ = > y {}^{2} = \frac{49}{\pi} \\ \\ = > y = \sqrt{ \frac{49}{\pi} } \\ \\ = > y = \frac{7}{ \sqrt{\pi} }$
Now radius = 2y

=> Radius =
$2 \times \frac{7}{ \sqrt{\pi} } \\ \\ = > \frac{14}{ \sqrt{\pi} }$


Now circumference = 2πr

=>
$2 \times \pi \times \frac{14}{ \sqrt{\pi} } \\ \\ = > 28 \sqrt{\pi}$


Answer is 28 √π

Option a

There should be 28 root pi though