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

More Triangle Problems

Category: Plane Geometry, Algebra

"Published in Suisun City, California, USA"

A right triangle has an area of 54 in². The product of the lengths of the three sides is 1620 in³. What are the lengths of its sides?

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

We know that the area of a right triangle is

If the product of the sides of a triangle is 1620 in³, then we can solve for the length of hypotenuse as follows

Use Pythagorean Theorem in order to solve for the other sides of a right triangle as follows

To solve for a and b, we need to use another equation which is 

or

Substitute the value of b to the above equation, we have

Multiply both sides of the equation by a², we have

Use Quadratic Formula in order to solve for a² as follows

If we choose the positive sign,

The other side of a right triangle is

If we choose the negative sign,

The other side of a right triangle is

Therefore, the three sides of a right triangle are 9, 12, and 15. 

Binomial Theorem, 2

Category: Algebra

"Published in Newark, California, USA"

Find the first three terms of 

Solution:

Consider the given equation above

Since the given polynomial is a trinomial, then we have to do something first in order to make this as a binomial and apply the Binomial Theorem to expand the binomial. 

To make a trinomial into a binomial, group x and -2y as follows

Since x and -2y are considered as one term in a group, then we can expand using the Binomial Theorem as follows

The above equation will be more complicated and longer if you will expand (x - 2y) completely. The more important is you know the principles of Binomial Theorem especially how to group the terms and how to expand the binomial which includes the getting the exponents of each terms as well as their coefficients. Therefore, the first three terms are

or