Question

EActivityIn this activity, you will compare the growth rates of linear, quadratic, and exponential functions in the contextof a real-world situation.Let's return to Mr. Smith's situation and help him choose one of the three retirement plans he's considering.We have the following three functions in three different forms:IMG_C

Answer 1

Part a:

Answer = the average amount by which the value of a retirement plan changed each year between year a and year b

Explanation: Each function represents the value of a retirement plan after a certain number of years. The average rate of change on the interval [a, b] is the total change in the value of a retirement plan between year a and year b divided by the number of years in the interval [a, b]. So, this value represents the average amount by which the value of a retirement plan changed each year between year a and year b.

Part b:

On the interval [0, 5], function f has the greatest average rate of change, and function h has the least average rate of change. On the interval [5, 10], function g has the greatest average rate of change, and function h has the least average rate of change.

Part c:

If Mr. Smith plans to retire in 20 years, then he should choose plan B

If he plans to wait longer than 20 years to retire, his best option is to

keep the same plan

Step-by-step explanation:

Evaluating the three functions for x = 20, function g has the greatest value.

f(20) = 26,000

g(20) = 36,500

h(20) = 24,403.02

So, Mr. Smith should choose plan B.

Since plan B is represented by an exponentially growing function, it has a continuously increasing rate of change. Plans A and C, which are modeled by a linear and a quadratic function, will ever exceed plan B once it has the greater value. After 20 years, plan B already has the greatest value. So, if he plans to wait longer than 20 years to retire, his best option is to keep the same plan.

Answer 2

Given:

The person plans to retire in, x = 20 years.

The objective is to select the correct plan and to choose the correct plant if the person decides to retire more than 20 years.

Step-by-step explanation:

Let's check the amount received at 20 years for three different plans.

To retire at 20 years;

In plan A:

At x = 20 in the given function,

`\begin{gathered} f(x)=800x+10000 \\ f(20)=800(20)+10000 \\ =16000+10000 \\ =26000 \end{gathered}`

Thus, the amount received from plant A is 26000.

In plan B:

From the given table, at x = 20 the amount received in plan B is 36500.

In plan C:

From the given quadratic function, at x = 20 the amount received in plan C is 24,403.2.

Since, by comparing all the three results, plan B obtains more amount compared to other plans.

To retire after 20 years:

If the person plans to retire longer than 20 years, then plan A gives a constant increase in amount after n number of years as it is a linear function.

But plan B is an increasing exponential function. So with increase in n number of years, the amount will also increase after n years.

Considering the graph of plan C, it appears to be a constant amount after successive 5 years interval.

Hence,

If the person retire at 20 years, he should choose plan B.

If the person plans to retire longer than 20 years, his best option is to keep the same plan.