Question

use the appropriate mean to solve the following:

1. a mutual fund increases 8% in 1 year, loses 5.2% in the next, and then is up 12.3% following year. what is the average percentage change?
2. a town grows from 12,456 people to 32,980 in 5 years.

a. what is the average total change per year?
b. what is the linear equation modeling this growth?
c. what is the average percentage change per year?
d. what is the exponential equation modeling this growth?

Explanation:

Problem 1

Step1: Identify the type of mean needed

For percentage changes in growth (like mutual fund returns), we use the geometric mean. First, convert the percentage changes to multipliers. A \(+8\%\) change is \(1 + 0.08=1.08\), a \(-5.2\%\) change is \(1 - 0.052 = 0.948\), and a \(+12.3\%\) change is \(1+0.123 = 1.123\).

Step2: Calculate the product of the multipliers

The product \(P=1.08\times0.948\times1.123\)

First, \(1.08\times0.948 = 1.08\times(1 - 0.052)=1.08-1.08\times0.052=1.08 - 0.05616 = 1.02384\)

Then, \(1.02384\times1.123=1.02384\times1+1.02384\times0.123=1.02384+0.12593232 = 1.14977232\)

Step3: Take the n - th root (n = 3, since there are 3 years)

The geometric mean \(GM=\sqrt[3]{1.14977232}\)

Using a calculator, \(\sqrt[3]{1.14977232}\approx1.047\)

Step4: Convert back to percentage change

The average percentage change is \((1.047 - 1)\times100\%=4.7\%\) (approx)

Step1: Calculate the total change in population

The initial population \(P_0 = 12456\), the final population \(P = 32980\), and the time \(t = 5\) years. The total change \(\Delta P=P - P_0=32980 - 12456 = 20524\)

Step2: Calculate the average total change per year

The average total change per year \(=\frac{\Delta P}{t}=\frac{20524}{5}=4104.8\)

Step1: Recall the linear equation form

The linear equation for population growth is \(P(t)=P_0+mt\), where \(P_0\) is the initial population, \(m\) is the slope (average rate of change), and \(t\) is time in years.

Step2: Substitute the values

We know \(P_0 = 12456\) and \(m = 4104.8\) (from part (a)). So the linear equation is \(P(t)=12456 + 4104.8t\)

Answer:

The average percentage change is approximately \(4.7\%\)

Problem 2

Part (a)