Similar Triangles

Category: Plane Geometry

"Published in Newark, California, USA"

The points C and D in the figure lie on level ground in the same vertical plane with the tip B of the tower AB. If the tower AB is 300 ft. high and measurements give A₁B₁ = 5 ft., CA₁ = 12 ft., A₂B₂ = 6 ft., and A₂D = 8 ft., find the distance CD.

Photo by Math Principles in Everyday Life

Solution:

The given problem above is asking for a distance of two points which is CD using similar triangles. The tower is located between CD and assuming that it is perpendicular to the ground. In this case, there are four right triangles in the figure. Label further the figure above, we have

Photo by Math Principles in Everyday Life

Consider ∆ABC:

Using similar triangles, we have

Consider ∆ABD:

Using similar triangles, we have

Therefore,

Variable Separation, 3

Category: Differential Equations, Integral Calculus, Algebra

"Published in Suisun City, California, USA"

Find the general solution for

Solution:

Consider the given equation above

The given equation above is a differential equation because it contains the differentials like dz and dt. We have to eliminate the differentials using the techniques of integration as follows

Using the separation of variables

Integrate on both sides of the equation

or

Multiply both sides of the equation by e^z, we have

More Integration Procedures, 9

Category: Integral Calculus, Trigonometry

"Published in Suisun City, California, USA"

Evaluate

Solution:

Consider the given equation above

There are two functions in the given equation which are x³ and sin x, respectively. If

 then

If

then

By applying the integration by parts, we have

Since the second term at the right side of the equation have two functions which are x² and cos x, then we have to apply the integration by parts again. If

then

If

then

Again, by applying the integration by parts, we have

Therefore, the first integration by parts becomes

Since the second term at the right side of the equation have two functions which are x and sin x, then we have to apply the integration by parts again. If

then

If

then

Again, by applying the integration by parts, we have

 

 

Therefore, the final answer is