Chapter 12.5, Problem 48SPR

To determine

Find the entire surface area of the bin with a conical top and bottom.

Answer to Problem 48SPR

  $1329.6f{t}^2$

Explanation of Solution

Given:

  

Formula Used: The Lateral area L of cone is

  $L=\pi rl$ , where l is the slant height and r is the radius of the base.

The Lateral area L of cylinder is

  $L=2\pi rh$ , where r is the radius and h is the height of the cylinder.

Calculation:

The bin is made up of cone on the top, cylinder in the middle and cone at the bottom.

Let the slant height of the top cone be l and the slant height of the bottom cone be m:

  

Diameter of the top and bottom cone is 18 ft. and height is 5 ft and 9 ft. respectively.

radius = r = $\frac{18}{2}=9ft.$

Use Pythagoras Theorem to find the slant height of the top cone = l = $\sqrt{5^2+9^2}=\sqrt{106}ft.$

Use Pythagoras Theorem to find the slant height of the bottom cone = m = $\sqrt{9^2+9^2}=9\sqrt{2}ft.$

Lateral area of the top cone = $L_{top}=\pi rl=\pi (9)(\sqrt{106})\approx 291.1f{t}^2$

Lateral area of the bottom cone = $L_{bottom}=\pi rm=\pi (9)(9\sqrt{2})\approx 359.9f{t}^2$

The middle cylinder has radius 9 ft. and height 12 ft.

Lateral area of the middle cylinder = $L_{middle}=2\pi rh=2\pi (9)(12)\approx 678.6f{t}^2$

So, the surface area of the bin = $L_{top}+L_{middle}+L_{bottom}\approx 291.1+259.9+678.6\approx 1329.6f{t}^2$

Hence,

Surface area of bin = $1329.6f{t}^2$