How Do You Find The Rate At Which A Population Grows?

Problems that ask you to solve for the rate r in exponential growth require the use of roots or creative use of exponents. Let’s look at an example.

Problem Suppose a town has a population of 5000 in 2010. If there are 7000 people in the town in 2012, what is the annual rate at which the population is growing?

Solution The easiest way to approach this problem is to think in terms of the exponential growth (compound interest formula),

This formula applies when population grows at an annual rate.

Let’s look at the quantities in the problem statement:

  • 5000 people originally > P = 5000
  • There is 7000 in the population after 2 years > A = 7000 and n = 2

Putting these values into the formula above gives us

We need to find the annual rate r. Since the r is hidden in the parentheses, we start by isolating the parentheses.

To get at the r, we need to remove the square on the parentheses.

Using a calculator to do the square root, we get r ≈ 0.183 or 18.3%.

Although most calculators have a square root key, when removing powers it is often useful to raise both sides to a power. For instance, we could remove the square by raising both sides to the ½ power:

When you raise a power to another power, you multiply the exponents 2 ∙ ½ = 1. The right side simply becomes 1 + r. Now we can solve for r:

Using the power key on your calculator gives the same answer as before. Make sure the 1/2 power is entirely in the power. You can make sure this happens using parentheses: (7000/5000)^(1/2)-1.

Now what if the population grows over six years instead of two years? Instead of a square on the parentheses we now have a sixth power.

To solve for r in this equation, we follow similar steps.

The root can be computed on a graphing calculator using the MATH button or put into WolframAlpha:

Either method gives r ≈ 0.577 or 5.77%. Notice that the annual rate is lower when it is earned over a longer period of time.

If we use a 1/6 power to solve for r, we would carry out the steps below:

Using a 1/6 power on your calculator gives the same answer as above.

How Do You Interpret A Linear Model?

Linear model are not always given in a nice slope-intercept form. If they were, it would be easy to read the slope and y-intercept off the y = mx + b form. In the example below, you are given a linear model in general form, ax + by = c. However, the letters are p and x instead of x and y.

Problem The percent p of high school seniors using marijuana daily can be related to x, the number of years after 1990, by the equation 30p – 19x = 30.

a. Find the x-intercept of the graph of this function.

Solution To find an intercept, we need to set the other variable equal to zero. In this case, to find the x-intercept, we need to set p = 0:

b. Find and interpret the p-intercept of the graph of this function.

Solution To find the p-intercept, set x = 0:

Since x corresponds to years after 1990, this tells us that in 1990 the percentage of seniors using marijuana daily was 1%.

c. Graph this function, using the intercepts. What values of x on the graph represent the years 1990 and after?

Solution Put the two points we have found on a graph.

Since 1990 and beyond correspond to x values 0 and above, a better viewing window would be one like we see below.

Notice that with each axis labeled, it makes more sense to use non-negative x values ( 0 and greater).

d. Use this model to determine the slope of the graph of this function if x is the independent variable.

Solution We could use the points we found earlier to calculate the slope, but it is easier to change the equation into slope-intercept form and read off the slope. To do this, we’ll need to solve 30p – 19x = 30 for p:

Since this is in the slope-intercept form p = mx + b, the slope is 19/30 or approximately 0.63. The line we drew above also makes sense since it has a positive slope and a vertical intercept of 1.

e. What is the rate of change of the percent of high school seniors using marijuana each year?

Solution The rate of change is the slope, with the appropriate units attached. In this case, the units are

So the percent of students using marijuana is rising at a rate of 19/30 percent per year. Since 19/30 is approximately .63, this means that each year the percentage increased by about .63%.


Interpreting this model means we examined the intercepts and their meaning as well as the slope and what it tells us about the percent using marijuana and the years since 1990.

How Do You Find the Equation of a Line From an Application?

Below are some problems that students solved on the board in previous semesters. All of them start from the slope-intercept form of a line and require you to find the slope m from two points and then solve for b. Click on the pictures to see a larger version.

Problem The percent p of adults who smoke cigarettes can be modeled by a linear equations p = mt + b, where t is the number of years after 1960. If two points on the graph of this function are (25, 30.7) and (50, 18.1), write the linear equation of this application.

Solution

 

Problem The number of women in the workforce, based on data and projections from 1950 to 2050, can be modeled by a linear equation y = mx + b. The number was 18.4 million in 1950 and is projected to be 81.6 million in 2030. Let x represent the number of years after 1950 and y be the number of women in the workforce in millions.

a. What is the slope of the line through (0, 18.4) and (80,81.6)?

b. What is the average rate of change in the number of women in the workforce during this time period?

c. Use the slope from part a and the number of million of women in the workforce in 1950 to write the equation of the line.

Solution

When Are Two Linear Models Equal?

In the problem below, you are given two lines. One corresponds to male enrollment in college and the other to female enrollment in college. The goal is to use these models to find when male enrollment equals female enrollment.

Problem Suppose the percent of males who enroll in college within 12 months of high school graduation is given by

y = -0.126x + 55.72

and the percent of females who enroll in college within 12 months of high school graduation is given by

y = 0.73x + 39.7,

where x is the number of years after 1960. Find the year these models indicate that the percent of females equaled the percent of males.

Click on either picture to see a larger version.

The x values in this problem correspond to years after 1960. Don’t be tempted to round the value to 19 and say the year is 1979. Since it asked in what year, 1960 + 18.71 is 1978.71 so the year they were equal is 1978.

How Do You Find Special Points on a Parabola?

Let’s look at how to use formulas for a parabola to get certain important points on a parabola.

Problem For the parabola y = 2x2 + 3x – 2, locate the points below.

a. The y-intercept.

Solution At the y-intercept, the x value is zero. This means that we need to set x = 0 in the equation:

y = 2(02) + 3(0) – 2 = -2

Putting this together, the y-intercept is at (0, -2).

b. The vertex.

Solution The vertex is located using the formula   where the values of a, b, and c come from the equation. In this case, a = 2, b = 3, and c = -2. This gives an x value on the intercept of

To find the corresponding y value, put this value into the equation,

This means the vertex is at (-3/4, –25/8).

c. The x-intercepts.

Solution At the x-intercepts, the y value is zero. Putting this into the equation yields

0 = 2x2 + 3x – 2

This equation is solved with the quadratic formula,

Put the values from the equation (a = 2, b = 3, and c = -2),

The x intercepts are at (-2, 0) and (1/2, 0).

All of these points are shown in the graph of the parabola below.