Spreadsheets have several built in functions for working with compound interest and annuities. To use these functions, we’ll start with a standard sheet.
This worksheet contains the variables used throughout Chapter 8. These variables correspond to these letter used in the text.
Number of periods is n
Annual interest rate is r
Payment is R
Present value is P
Future value is A
Periods per year is m
Values given in a problem will be entered in column B. Values calculated by the spreadsheet will be entered in column C. We will also assume that amounts paid out are negative and amounts received are positive.
In another MathFAQ, I examined how we can find the equation of a line from two data points. In this post I want to look at a closely related problem where we find the equation of the line from a rate.
Problem Assume the growth of the population of Del Webb’s Sun City Hilton Head community was linear from 1996 to 2000, with a population of 198 in 1996 and a rate of growth of 705 people per year.
a. Write an equation for the population P of the community where x is the number of years after 1990.
Solution The population of 198 in 1996 corresponds to the point (6, 198) since the variable x corresponds to years after 1990. We’ll write the slope-intercept form of the line, P = mx + b, and substitute m into the equation. The rate of growth, 705 people per year, is the slope of the function. Therefore, the line describing the population is
P = 705x + b
To find the value of b, we need to substitute x = 6 and P = 198:
198 = 705(6) + b
-4032 = b
This gives us the equation,
P = 705x – 4032
b. Use the function to estimate the population in 2002.
Solution The year 2002 corresponds to x = 12. Substitute this value into the function to yield
P = 705(12) – 4032 = 4428
The population in 2002 will be 4428 people.
c. In what year will the population reach 10,000?
Solution In this part, set P = 10,000 and solve for x.
10000 = 705x – 4032
14032 = 705x
14032/705 = x
19.9 ≈ x
This corresponds to 1990 + 19.9 = 2009.9. So in the year 2009, the population will reach 10000. Since we are asking in what year, we DO NOT round up on the answer.
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.
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.
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.
In 
