# Frustum

## 041 Review Problem - Weight capacity of industrial soap kettle

**Problem 41**

Soap kettles used in the commercial manufacture of soap are as a rule large cylindrical vats, 50,000 lb. or more of soap being made in a single beating. Find the capacity of such a kettle having an inside diameter of 18 ft and an altitude of 30 ft. if soap weighs 70 lb. per cu. ft.

## 039 Review Problem - Bushels of wheat the grain elevator can hold

**Problem 39**

A grain elevator in the form of a frustum of a right circular cone is 24 ft. high, and the radii of its bases are 10 ft. and 5 ft., respectively; how many bushels of wheat will it hold if 1-1/4 cu. ft. equals 1 bu.?

## 032 Review Problem - How many cups of coffee a coffee pot can hold?

**Problem 32**

A coffee pot is 5 in. deep, 4-1/2 in. in diameter at the top, and 5-3/4 in. in diameter at the bottom. How many cups of coffee will it hold if 6 cups equal 1 quart? Answer to the nearest whole number.

## 025 Review Problem - Time required to fill a reservoir of water

**Problem 25**

A reservoir 10 ft. deep is in the form of a frustum of inverted square pyramid with bases of 100 and 90 ft. on a side respectively. How long will it require an inlet pipe to fill the reservoir if the water pours in at the rate of 200 gal. per min.? (One gal. = 231 cu. in.)

## 022 Review Problem - Tin required to create a funnel

**Problem 22**

How many square feet of tin are required to make a funnel, if the diameters of the top and bottom are 28 in. and 14 in., respectively, and the height is 24 in.?

## 013 Review Problem - Volume of water inside the Venturi meter

**Problem 13**

The accompanying figure represents the longitudinal view of a Venturi meter, a device designed to measure the flow of water in pipes. If the throat of the of the meter is 6 in. long and has an inside diameter of 4 in., find the volume of water in the meter which is used in 12-in. pipe line if the altitudes of the tapering parts are in the ratio 1:3 and the smaller altitude measures 12 in.

## Frustums

*Frustum* of a pyramid (or cone) is a portion of pyramid (or cone) included between the base and the section parallel to the base not passing through the vertex.

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## Derivation of Formula for Lateral Area of Frustum of a Right Circular Cone

The lateral area of frustum of a right circular cone is given by the formula

where

R = radius of the lower base

r = radius of the upper base

L = length of lateral side

## Derivation of formula for volume of a frustum of pyramid/cone

**Frustum of a pyramid and frustum of a cone**

The formula for frustum of a pyramid or frustum of a cone is given by

Where:

h = perpendicular distance between A_{1} and A_{2} (h is called the altitude of the frustum)

A_{1} = area of the lower base

A_{2} = area of the upper base

Note that A_{1} and A_{2} are parallel to each other.