Saturday, February 28, 2015

Rectangular Parallelepiped Problems, 17

Category: Solid Geometry

"Published in Newark, California, USA"

Find the angles that the diagonal of a rectangular parallelepiped 1 in. by 3 in. by 5 in. make with the faces.

Solution:

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

 Photo by Math Principles in Everyday Life

Next, we need to draw a diagonal line at the top and bottom of the rectangular parallelepiped as well as its diagonal and angles as follows

 Photo by Math Principles in Everyday Life

By using Pythagorean Theorem, the length of d is

Therefore, the value of an angle is

or

Next, we need to draw a diagonal line at the left and right of the rectangular parallelepiped as well as its diagonal and angles as follows

 Photo by Math Principles in Everyday Life

Therefore, the value of an angle is

or

Next, we need to draw a diagonal line at the front and back of the rectangular parallelepiped as well as its diagonal and angles as follows

 Photo by Math Principles in Everyday Life

Therefore, the value of an angle is

or

Friday, February 27, 2015

Trapezoid Prism Problems, 8

Category: Solid Geometry

"Published in Newark, California, USA"

The figure shows a right section of a railroad cut in a hillside. If the sides rise 1 unit vertically to a horizontal distance of 1.5 units, and if the length of the cut is 100 ft., find the volume of earth removed.

 Photo by Math Principles in Everyday Life

Solution:

The cross section of a railroad cut in a hillside is a general quadrilateral. If you draw vertical lines from the top ends of a general quadrilateral to the bottom horizontal line that contains one side, then it becomes a trapezoid as follows

 Photo by Math Principles in Everyday Life

By similar triangles, we can solve for the values of a and b as follows

The area of a trapezoid is

The area of small triangle is

The area of big triangle is

Hence, the area of the base which is the cross section of a railroad cut in a hillside is

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

Thursday, February 26, 2015

Square Prism Problems, 4

Category: Solid Geometry

"Published in Vacaville, California, USA"

The figure represents a truncated prism. The base ABCD is a square, and the lateral edges AE, BF, CG, DH are perpendicular to the base. AB = 10 in., AE = 6 in., BF = 10 in., CG = 10 in., DH = 6 in. Find (a) the length of each face diagonal; (b) each face angle; (c) the length of each diagonal of solid; (d) the lateral area; (e) the total area; (f) the angle BFH; (g) the volume.

 Photo by Math Principles in Everyday Life

Solution:

(a) Label further the figure and draw a diagonal line at each face as follows

 Photo by Math Principles in Everyday Life

By Pythagorean Theorem, the length of the diagonals are

(b) Let's consider the side view of a truncated prism which is HDCG as follows

 Photo by Math Principles in Everyday Life

By using basic trigonometric functions, we can solve for the face angles as follows

or

or

The opposite side of a truncated prism which is AEFB, their face angles are also 68°11'55'' and 111°48'5''. The rest of the face angles are all 90°

(c) Let's consider the given figure again and draw their diagonal lines as follows

 Photo by Math Principles in Everyday Life

The length of the diagonals of a truncated prism are

(d) The lateral area of a truncated prism is

(e) The total area of a truncated prism is

(f) Let's consider the given figure again as follows

 Photo by Math Principles in Everyday Life

By Cosine Law, the value of ∠BFH which is also Î± is

or

(g) The volume of a truncated prism is