Question

Coffee is sold in two different-sized canisters. The smaller canister has a diameter of 9 cm and a height of 12 cm. The larger canister is double the size of the small canister. Calculate the volume and surface area of each canister and compare the results of doubling the dimensions.

Answer 1

Explanation:

Smaller Canister:

Height, h = 12cm

Diameter = 9cm

Radius, r = 4.5 cm

`Volume = \pi r^2h\\`

`=\pi * 4.5^2 * 12\\\\=243 \pi \ cm^3`

`Surface \ area = 2 \pi r(r + h)\\`

`=2 * \pi * 4.5 ( 4.5 + 12)\\\\= \pi * 9(16.5)\\\\=148.5 \ cm^2`

Larger Canister:

Measures double the smaller canister, that is

height, H = 24 cm

Diameter = 18cm

Radius, R = 9cm

`Volume = \pi R^2 H\\`

`= \pi * 9^2 * 24\\\\= \pi * 81 * 24 \\\\= 1944 \pi`

`Surface \ area = 2 \pi R(R+ H)`

`= \pi * 2 * 9 ( 9 + 24) \\\\= \pi * 18(33)\\\\=594 \pi \ cm^2`

Comparing results :

`Volume_(small) = \pi r^2 h\\\\Volume_(large) = \pi R^2 H = \pi(2r)^2(2h) = \pi(4r^2)(2h)= 8 \pi r^2h = 8 * volume_(small)`

Therefore, volume of larger canister is 8 times the volume of smaller canister.

`Surface\ area _(larger) = 2 \pi R(R + H) = 2 \pi(2r)((2r+2h))\\\\`

`=2 \ pi * 2r * 2 (r+ h)\\\\= 4 * 2\pi r(r+ h) \\\\= 4 * surface\ area_(smaller)`

Therefore, surface area of larger canister is 4 times the surface area of smaller canister.