Chapter 12.3, Problem 13PSC

To determine

To calculate: The total area of a cone frustum if slant height is 5.

Answer to Problem 13PSC

The total area of frustum is $140\pi$ .

Explanation of Solution

Given information:

Slant height of frustum cone = 5,

Radius of top base = 4,

Radius of bottom base = 8.

Formula used:

Area of cone $=\frac{1}{2}\times C\times l$

C = Circumference of cone and l = slant height of cone

r = radius of circle

Area of a circle: $A=\pi r^2$

r = radius of circle

Calculation:

  

Radius of larger circle is 8 and radius of smaller circle is 4.

The converse of Midline theorem is used. Midline theorem states that the segment that joins the midpoints of two sides of a triangle is parallel to the third side and half as long

Slant height of top cone = 10.

Slant height of bottom cone = 5.

Circumference of top cone: $C=2\pi r=2\pi \times 4=8\pi$

Lateral Area of top cone $=\frac{1}{2}\times C\times l$

Lateral Area of top cone $=\frac{1}{2}\times 8\pi \times 5$

Lateral Area of top cone $=20\pi$

Circumference of bottom cone: $C=2\pi r=2\pi \times 8=16\pi$

Lateral Area of bottom cone $=\frac{1}{2}\times C\times l$

Lateral Area of bottom cone $=\frac{1}{2}\times 16\pi \times 10$

Lateral Area of bottom cone $=80\pi$

Difference in Lateral Area $=80\pi -20\pi =60\pi$

Area of top base circle $=\pi {\left(8\right)}^2=64\pi$

Area of bottom base circle $=\pi {\left(4\right)}^2=16\pi$

Total Area = Area of bottom base circle + Area of top base circle + Difference in Lateral Area

Total Area $=16\pi +64\pi +(80\pi -20\pi )$

Total Area $=80\pi +60\pi$

Total Area $=140\pi$