"Published in Newark, California, USA"
A block of granite is in the form of the frustum of a regular square pyramid whose upper and lower base edges are 3 ft. and 7 ft., respectively. If each of the lateral faces is inclined at an angle of 62°30' to the base, find the volume of granite in the block.
Solution:
To illustrate the problem, it is better to draw the figure as follows
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| Photo by Math Principles in Everyday Life |
Since the length of the edges of the upper and lower bases are given, then we can solve for the area of the bases. For the upper base, the area is
and for the lower base, the area is
The altitude of the frustum of a regular square pyramid is not given in the problem. If the angle of inclination of each lateral faces to the lower base is given, then we can solve for the altitude by isolating and label further the vertical section of the frustum of a regular square pyramid as follows
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| Photo by Math Principles in Everyday Life |
There are two equal right triangles in the vertical section of the frustum of a regular square pyramid. This section is an isosceles trapezoid. By using simple trigonometric function, the altitude of the section which is also the altitude of the frustum of a regular square pyramid is
Therefore, the volume of the frustum of a regular square pyramid which is the volume of a block of granite is
