Question

A man makes complete use of 405 cc of iron, 783 cc of aluminium, and 351 cc of copper to make a number of solid right circular cylinders of each type of metal. These cylinders have the same volume and each of these has radius 3 cm. If the total number of cylinders is to be kept at a minimum, then the total surface area of all these cylinders, in sq cm, is

• A. \(8464\pi\)
• B. \(928\pi\)
• C. \(1044(4+\pi)\)
• D. \(1026(1+\pi)\)

Answer 1

The Correct Option is D

To solve the problem, we need to determine the volume and number of cylinders that can be made from each type of metal and ensure that they are minimized. Each cylinder has a radius \( r = 3 \) cm. The formula for the volume of a cylinder is \( V = \pi r^2 h \). Given:

• Total volume of iron = \(405 \text{ cc}\)
• Total volume of aluminium = \(783 \text{ cc}\) 
• Total volume of copper = \(351 \text{ cc}\)

The cylinders have equal volumes, thus the height \( h \) is the same for all metals. Setting the volume formula equal to each metal volume yields:

• \( \pi \times 3^2 \times h = 405 \)
• \( \pi \times 3^2 \times h = 783 \)
• \( \pi \times 3^2 \times h = 351 \)

We get:

• For iron, \( h = \frac{405}{9\pi} = \frac{45}{\pi} \)
• For aluminium, \( h = \frac{783}{9\pi} = \frac{87}{\pi} \)
• For copper, \( h = \frac{351}{9\pi} = \frac{39}{\pi} \)

We need a common height to ensure each cylinder has the same volume across different metals. Find the GCD of the volumes for determining the minimum number of cylinders:

• GCD of \(405, 783, 351\) is \(27\)

So, the volume of each cylinder = \(27\) cc. Find total number of cylinders:

• Total cylinders = \( \frac{405+783+351}{27} = 57 \)

Each cylinder has radius \( r = 3 \) cm, height = \( \frac{27}{9\pi} = \frac{3}{\pi} \). Calculate the surface area of a cylinder:

• Total surface area of cylinder = \(2\pi rh + 2\pi r^2 = 2\pi \times 3 \times \frac{3}{\pi} + 2\pi \times 3^2 = 6 + 18\pi\)

Thus, the total surface area for all cylinders is:

• Total surface area = \( 57 \times (6 + 18\pi) = 342 + 1026\pi\)

This results in the total surface area being \(1026(1+\pi)\) square cm, making it the correct option.