Math Principles

Sunday, August 17, 2014

Triangular Prism Problems, 4

Category: Solid Geometry

"Published in Vacaville, California, USA"

The trough shown in the figure has triangular ends which lie in parallel planes. The top of the trough is a horizontal rectangle 20 in. by 33 in., and the depth of the trough is 16 in.

(a) How many gallons of water will it hold? (One gal. = 231 cu. in.)
(b) How many gallons does it contain when the depth of the water is 10 in.?
(c) What is the depth of the water when the trough contains 3 gals.?
(d) Find the wetted surface when the depth of the water is 9 in.

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Solution:

(a) Consider the given figure above

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The area of the base is

Therefore, the volume of a trough when filled with water which is the volume of a prism is

The volume of a trough in gallons is

(b) Consider the front side of a trough

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Bu using similar triangles, the area of the base with water is

Therefore, the volume of a trough with water is

The volume of a trough with water in gallons is

The area of the base of a trough with water is

By using similar triangles

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Therefore, the depth of water in a trough is

(d) Consider the front side of a trough

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By using similar triangles, the length of the base is

Let's assume that the base of a trough is an isosceles triangle.

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By using Pythagorean Theorem, the length of the wetted edge is

The area of the wetted base of a trough is

The lateral area of the wetted surface of a trough is

Therefore, the total area of the wetted surface of a trough is

Saturday, August 16, 2014

More Prism Problems, 2

Category: Solid Geometry

"Published in Newark, California, USA"

A railway cut 200 ft. and 30 ft. wide at the bottom is made with sloping sides, which are 80 ft. and 60 ft. in length. The 80 ft. side is inclined 45° and and the other side is inclined 30° to the horizontal. Find the cost of removing the earth at $2 per load, if the trucks have a capacity of 4 cu. yd.

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Solution:

Consider the given figure above

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To understand more the problem, it is better to analyze the front side which is a quadrilateral and the base of the prism, and then label further as follows

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If you draw vertical lines from the two upper vertices and horizontal lines from the two lower vertices of a quadrilateral, then it becomes a trapezoid. Since the sides and angles of a quadrilateral are given, then we can solve the remaining sides of two right triangles.