Bottles of perfume sometimes have thick glass (n = 1.5) walls which give the impression the volume is larger than it really is. Assume a cylindrical bottle with an inner radius of 1.0 cm and an outer radius of 2.0 cm. What percentage of the apparent volume is the real volume?
A) 95%
B) 60%
C) 80%
D) 64%
E) 75%

Question

Quizplus · Accepted Answer

The apparent volume includes the thick glass walls, so we need to find the difference between the volume of the cylinder with outer radius 2.0 cm and the volume of the cylinder with inner radius 1.0 cm.

The volume of the cylinder with outer radius 2.0 cm is V1 = πr^2h = π(2.0 cm)^2h = 4πh cm^3. 
The volume of the cylinder with inner radius 1.0 cm is V2 = πr^2h = π(1.0 cm)^2h = πh cm^3.

The difference in volume is V1 - V2 = (4π - π)h = 3πh cm^3.

The real volume is the volume without the thick glass walls, so it is just the volume of the cylinder with inner radius 1.0 cm, which is V2 = πh cm^3.

To find the percentage of the apparent volume that is the real volume, we divide the real volume by the apparent volume and multiply by 100%.

(real volume / apparent volume) x 100% = (V2 / V1) x 100% 
= (πh / 4πh) x 100% 
= 25%

Therefore, the real volume is 25% of the apparent volume. To find the percentage of the apparent volume that is not the real volume, we subtract from 100%:

100% - 25% = 75%

So 75% of the apparent volume is the thick glass walls, and only 25% is the actual perfume inside.

The closest answer choice is (D) 64%, which is not exactly correct, but the necessary calculations were not provided to produce that value.

Bottles of Perfume Sometimes Have Thick Glass (N = 1