   Chapter 9.2, Problem 33E Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Solutions

Chapter
Section Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

The street department’s storage building, which is used to store the rock, gravel, and salt used on the city’s roadways, is built in the shape of a regular hexagonal pyramid. The altitude of the pyramid has the same length as any side of the base. If the volume of the interior is 11 , 972 f t 3 , find the length of the altitude and of each side of the base to the nearest foot.

To determine

To find:

The length of the altitude and of each side of the base to the nearest foot.

Explanation

Given:

A regular hexagonal pyramid and the altitude of the pyramid has the same length as any side of the base i.e. h=x. The volume of the pyramid is 11,972ft3.

Properties Used:

A pyramid is made by connecting a base to an apex. The base is flat with straight edges, no curves, hence, a polygon and all other faces are triangles.

A regular pyramid is a pyramid whose base is a regular polygon and whose lateral edges are all congruent.

The lateral area L of a regular pyramid with slant height of length l and perimeter P of the base is given by

L=12lP.

The total surface area T of a pyramid with lateral area L and base area B is given by

T=L+B.

According to the Pythagorean theorem, in a right-angled triangle

hypotenuse2=base2+perpendicular2.

In a regular pyramid, the lengths of the apothem a of the base, the altitude h, and the slant height l satisfy the Pythagorean Theorem; that is, l2=a2+h2.

The perimeter of the square is four times the side.

The volume V of a pyramid having a base area B and an altitude of length h is given by V=13Bh.

Calculation:

Since, the sides of the hexagonal base measure x each and the angle at the center for each base is 60° which is bisected by the perpendicular so the apothem in the right-angled triangle is given by the perpendicular side i.e. a=side2tan30°

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