Trapezoid Prism Problems, 4

Category: Solid Geometry

"Published in Newark, California, USA"

A dam 100 ft. long has a cross section which is a trapezoid whose altitude is 16 ft., and whose upper base is 5 ft. If the lower base angles of the cross section are 50° and 65°, find the volume of material the dam contains.

Solution:

To illustrate the problem, it is better to draw the figure as follows

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Since the length of the lower base is not given, then we have to isolate the base first and then label further so that we can solve for its length.

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Consider the 50° right triangle,

 
Consider the 65° right triangle,

The length of the lower base of a trapezoid which is the cross section of a dam is

 

 

Hence, the area of the base is

Therefore, the volume of a trapezoid prism which is a dam is

Triangular Prism Problems, 5

Category: Solid Geometry

"Published in Newark, California, USA"

The Pennsylvania Railroad found it necessary, owing to land slides upon the roadbed, to reduce the angle of inclination of one bank of a certain railway cut near Pittsburgh, PA, from an original angle of 40° to a new angle of 25°. The bank as it originally stood was 200 ft. long and had a slant length of 60 ft. Find the amount of the earth removed, if the top level of the bank remained unchanged. 

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Solution:

Consider the given figure above

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From the two upper vertices of a triangle, draw vertical lines perpendicular to the ground and then label further the figure as follows

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Consider a 40° right triangle,

The value of h in a 25° right triangle is equal to the value of h in a 40° right triangle.

Consider a 25° right triangle,

The length of the base of an obtuse triangle is

The area of an obtuse triangle which is the base of the prism is

Therefore, the amount of earth removed which is the volume of a prism is

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

(c) The volume of a trough with water in cubic inches 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