Right Circular Cylinder - Sphere

Category: Solid Geometry

"Published in Newark, California, USA"

Two balls, one 6 in. in diameter and the other 4 in. in diameter are placed in a cylindrical jar 9 in. in diameter, as shown. Find the volume of water necessary to cover them.

Photo by Math Principles in Everyday Life

Solution:

The first that we have to do is to analyze and label the given figure as follows

Photo by Math Principles in Everyday Life

Consider a right triangle between the centers of two spheres and use Pythagorean Theorem in order to solve for x as follows

The height of a right circular cylinder is

The volume of a right circular cylinder is

The volume of a big sphere is

The volume of a small sphere is

Therefore, the volume of water necessary to cover the two balls is

or you can give the value of pi as follows

Derivative - Trigonometric Functions

Category: Differential Calculus, Trigonometry

"Published in Suisun City, California, USA"

If n is a positive integer, prove that

Solution:

Consider the given equation above

We will use the left side of the equation to prove the right side of the equation as follows

Apply the derivative by product and then trigonometric functions, we have

Separate their common factor, we have

Therefore,

Exponential Growth - Population Problem

Category: Algebra

"Published in Suisun City, California, USA"

The population of California was 10,586,223 in 1950 and 23,668,562 in 1980. Assume the population grows exponentially.

(a) Find a formula for the population t years after 1950.
(b) Find the time required for the population to double.
(c) Use the data to predict the present population of California.

Solution:

The given problem above is about the exponential growth for a population in California. The exponential growth is given by the formula

where

     x = population at time t
   x₀ = initial size of population
     r = relative rate of growth (expressed as a proportion of the population)
     t = time of growth

Now, if the population of California in 1950 (initial time) is 10,586,223 and in 1980 (final time) is 23,668,562, the growth rate r will be equal to

Take natural logarithm on both sides of the equation, we have

Therefore, the population of California in time t is

After 1950, the population of California will be doubled for

Take natural logarithm on both sides of the equation, we have

or

Therefore, the population of California will be doubled in 1950 + 26 = 1976.

In 2013, the population of California is

or