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

Rate, Distance, Time - Problem, 3

Category: Algebra

"Published in Newark, California, USA"

A jeepney enroute to Bataan travels for 1 hour and 40 minutes at a uniform speed. Inclement weather forces the driver to slow down to two-fifths of the original speed. The jeepney reaches its destination 30 minutes later covering a total distance of 84 miles. What was the original speed?

Solution:

The given word problem is about the rate, distance, and time problem. Let's analyze the word problem as follows

Let     x = be the original speed of the jeepney before the inclement weather
     ²/₅ x = be the speed of the jeepney during the inclement weather

To understand more the word problem, it is better to illustrate the given word problem as follows

We know that

The total distance traveled by the jeepney is equal to the distance traveled at a uniform speed plus the distance traveled at a reduced speed during the inclement weather. The working equation for this problem is 

Therefore, the original speed of the jeepney is 45 miles/hr.