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Tuesday, April 30, 2013

Subtended Angle Problem

Category: Trigonometry

"Published in Newark, California, USA"

A vertical pole is 35 ft. high and stands on sloping ground. It is braced by a guy rope which extends from the top of the pole to a point on the ground 25 ft. from the foot of the pole. The pole subtends an angle of 30° at the point where the guy rope reaches the ground. Determine the length of the rope.

Solution:

The given word problem is about subtended angle. Subtended angle is the angle formed by an object at a given external point. Slopping ground means that the ground is not horizontal and it is slanted or inclined. To illustrate the problem, it is better to draw the figure as follows


Photo by Math Principles in Everyday Life

The above figure forms an oblique triangle. Analyze further the figure, we have


Photo by Math Principles in Everyday Life

From the given items of an oblique triangle, use Sine Law in order to solve for angle A as follows













We know that the sum of the angles of a triangle is always equal to 180°. If we know the two angles of a triangle, then we can solve for the other angle as follows









Finally, use Sine Law again in order to solve for the length of a rope as follows












Monday, April 29, 2013

Word Problem - Age Problem

Category: Algebra

"Published in Newark, California, USA"

Six years ago, Rosario was 4 times as old as her daughter. Ten years from now, she will be only twice as old as her 
daughter. How old are they now?

Solution:

The given word problem is about age problem where you   need to solve for the age of a person or people. Let's analyze the given word problem above as follows:

Let         x = be the present age of Rosario
              y = be the present age of Rosario's daughter
         x - 6 = be the age of Rosario six years ago
         y - 6 = be the age of Rosario's daughter six years ago
      x + 10 = be the age of Rosario ten years from now
      y + 10 = be the age of Rosario's daughter ten years from now.

If the first statement says, "Six years ago, Rosario was 4 times as old as her daughter." then the first working equation will be









If the second statement says, "Ten years from now, she will be only twice as old as her daughter." then the second working equation will be









Next, equate the two working equations in order to solve the age of Rosario's daughter as follows











Substitute the value of y to either of the two working equations in order to solve for the age of Rosario as follows









Therefore, in present time, Rosario is 38 years old while her daughter is 14 years old.


Sunday, April 28, 2013

More Integration Procedures, 7

Category: Integral Calculus

"Published in Suisun City, California, USA"

Evaluate



Solution:

Consider the give equation above



If
then

Therefore, the given equation above can be integrated by simple integration using integration by power as follows


















Saturday, April 27, 2013

Solving Trigonometric Equations, 4

Category: Trigonometry, Algebra

"Published in Suisun City, California, USA"

Find the value of x for 



Solution:

Consider the given equation above



By looking at the equation above, the factors of 4sin x cos x are 2sin x and 2cos x while the factors of -1 are 1 and -1. Since there are 2sin x, -2cos x, and -1 in the equation above, then we can rewrite the above equation as follows



Equate each factor to zero and solve for the value of x.

If
then







Since the value of sine is positive at 1st and 2nd Quadrant, then 





where n is the number of revolutions.

If
then







Since the value of cosine is negative at 2nd and 3rd Quadrant, then 





where n is the number of revolutions.