Derive the formula for the curved surface area and total surface area of the frustum of a cone, given to you in Section 13.5, using the symbols as explained.

Sure, let's derive the formula for the curved surface area and total surface area of the frustum of a cone.

First, let's define the symbols:

1. h1 = height of the larger cone
2. h2 = height of the smaller cone that is cut off
3. h = height of the frustum (h = h1 - h2)
4. r1 = radius of the base of the larger cone
5. r2 = radius of the base of the smaller cone
6. l1 = slant height of the larger cone
7. l2 = slant height of the smaller cone
8. l = slant height of the frustum

Curved Surface Area (CSA) of a cone is given by πrl. So, the CSA of the larger cone is πr1l1 and the CSA of the smaller cone is πr2l2.

The CSA of the frustum is the difference between the CSA of the larger cone and the smaller cone.

So, CSA of frustum = πr1l1 - πr2l2

Now, let's derive the formula for the Total Surface Area (TSA) of the frustum.

The TSA of a cone is given by CSA + area of base. So, the TSA of the larger cone is πr1l1 + πr1^2 and the TSA of the smaller cone is πr2l2 + πr2^2.

The TSA of the frustum is the difference between the TSA of the larger cone and the smaller cone, plus the area of the base of the frustum (πr2^2).

So, TSA of frustum = (πr1l1 + πr1^2) - (πr2l2 + πr2^2) + πr2^2 = πr1l1 - πr2l2 + πr1^2 - πr2^2 + πr2^2 = CSA of frustum + πr1^2

So, the formulas for the CSA and TSA of the frustum of a cone are:

CSA = πr1l1 - πr2l2 TSA = CSA + πr1^2