Eliminating Arbitrary Constant

Category: Differential Equations, Algebra

"Published in Newark, California, USA"

Eliminate the arbitrary constant for

Solution:

Consider the given equation above and call this as equation #1, we have

Take the derivative with respect to x and call this as equation #2, we have

Since there are two arbitrary constants in the given equation, then we have to take the derivative of the given equation twice with respect to x. So, take the derivative of the above equation again with respect to x and call this as equation #3, we have

Using the equations #1 and #2:

Multiply equation #1 by 2 and multiply equation #2 by -1. Add the two equations in order to eliminate c₁ and call this as equation #4, we have

                                —————————————————

Using the equations #1 and #3:

Multiply equation #1 by 4 and multiply equation #3 by -1. Add the two equations in order to eliminate c₁ and call this as equation #5, we have

                                —————————————————

Equate equations #4 and #5, we have

Multiply both sides of the equation by 5, we have

Therefore,

Volume - Cube, Given Diagonal, 2

Category: Solid Geometry

"Published in Newark, California, USA"

If a cube has an edge equal to the diagonal of another cube, find the ratio of their volumes.

Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above

Photo by Math Principles in Everyday Life

Let V₁ = be the volume of a small cube
        y = be the edge of a small cube
      V₂ = be the volume of a big cube
        x = be the edge of a big cube

For a small cube:

If the diagonal of a small cube is x, then the value of the edges, y will be equal to

Therefore, the volume of a small cube is

For a big cube:

The volume of a big cube is 

Therefore, the ratio of their volumes is equal to

Proving Trigonometric Identities, 4

Category: Trigonometry

"Published in Newark, California, USA"

If A + B + C = 180°, prove that

Solution:

Consider the given equation above

In proving of trigonometric identities, we have to choose first the more complicated part and then simplify until it match with the other side of the equation. In this case, we have to choose the right side of the equation as follows

Use the Sum and Product Formula, we have

but

Therefore, the above equation becomes

Therefore, the identity is correct for