Question

A lampshade has a height of 12cm and upper and lower diameters of 10cm and 20cm A)what area of material is required to cover that curved surface of the frustum B)what is the volume of the frustum (Give both answers in terms of pie)

Answer 1

`(a)\ Area = 195\pi`

`(b)\ Volume = 700\pi`

Explanation:

Given

`h = 12`

`d_1 = 10` --- lower diameter

`d_2 = 20` --- upper diameter

Solving (a): The curved surface area

This is calculated as:

`Area = \pi l(r_1 + r_2)`

Where

`r_1 = 0.5 * d_1 = 0.5 * 10 = 5` --- lower radius

`r_2 = 0.5 * d_2 = 0.5 * 20 = 10` --- upper radius

And

`l = \sqrt{h^2 + (r_1 - r_2)^2` ---- l represents the slant height of the frustrum

`l = \sqrt{12^2 + (5 - 10)^2`

`l = \sqrt{12^2 + (-5)^2`

`l = \sqrt{144 + 25`

`l = \sqrt{169`

`l = 13`

So, we have:

`Area = \pi l(r_1 + r_2)`

`Area = \pi * 13(5 + 10)`

`Area = \pi * 13(15)`

`Area = 195\pi`

Solving (b): The volume

This is calculated as:

`Volume = (1)/(3)\pi * h * (r_1^2 + r_2^2 + (r_1 * r_2))`

This gives:

`Volume = (1)/(3)\pi * 12 * (5^2 + 10^2 + (5 * 10))`

`Volume = (1)/(3)\pi * 12 * (25 + 100 + (50))`

`Volume = (1)/(3)\pi * 12 * (175)`

`Volume = \pi *4 * 175`

`Volume = 700\pi`