Triangular Prism Problem

Category: Solid Geometry, Trigonometry

"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 45º to a new angle of 30º. 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. 

Photo by Math Principles in Everyday Life

Solution:

The given figure is the cross section of a bank. The darker area of the cross section will be removed so that the inclination will be 30º instead of 45º. The darker area is also an obtuse triangle and we need to find the area of that one. Let's start to get the sides of the triangle by analyzing the figure as follows

Photo by Math Principles in Everyday Life

Consider the 45º triangle in order to solve for h and c as follows

Consider the 30º triangle in order to solve for d as follows

The base of a triangle can be calculated as follows

The area of a triangle is also the area of the base is calculated as follows

Finally, the volume of a prism is also the amount of the soil or earth removed from the bank is calculated as follows

Integration - Miscellaneous Substitution

Category: Integral Calculus, Trigonometry

"Published in Newark, California, USA"

Find the integral for

Solution:

You notice that the denominator contains trigonometric functions and we cannot integrate it by simple integration. This is a difficult one because the numerator has no trigonometric functions. If you will use the integration by parts, then the above equation will be more complicated and there will be an endless repetition of the procedure. 

For this type of a function, like the given equation above, we can integrate it by Miscellaneous Substitution. Let's proceed with the integration technique as follows

Let 

From the double angle formula,

Since the given problem has Sine and Cosine functions, then we can get the values of Sine and Cosine functions from Tangent function as follows

Using the figure above that

From the given problem

Substitute the values of dx, Sin x, and Cos x to the above equation, we have

but

Therefore,

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Converting Decimal Number - Fraction

Category: Arithmetic

"Published in Newark, California, USA"

Express 0. 0125 into its equivalent fraction. 

Solution:

The above decimal is known as a common decimal or terminating decimal. Terminating decimal means that the decimal digits are not repeating endlessly and there's an end in division if you divide the numerator by the denominator by manual division. If the denominator is a factor of 2, 5, and 10, then the quotient is a terminating decimal. On the other hand, if the denominator is not a factor of 2, 5, and 10, then the quotient is a repeating decimal. On December 4, 2012, we discussed about the conversion of a repeating decimal into a fraction.

How do you convert a terminating decimal into a fraction? Well, let's consider this procedure and solution

Let                x = 0.0125

         10,000 x = 125

(We multiply both sides of the equation by 10,000 because we want the right side of the equation to move the decimal point to the right. One decimal movement is equivalent to a power of 10. If you move a decimal position to the right, then you multiply a number by 10. If you move two decimal position to the right, then you multiply a number by 100, and so on.)

Divide both sides by 10,000

As a rule in Mathematics that all fractions must be simplified into a lowest term. Factor 125 and 10,000 as follows

Cancel three 5's as their Greatest Common Factor (GCF) in a fraction

Therefore, the answer is