We use exponents when exploring real world situations.
Early Exponentials
An athlete signs a contract saying that they will earn $8.3 million with an increase of
4.8% each year of the 5 year contract. Use a calculator to fill in the following table.
Round to the nearest tenth of a million:
Here we see how we can simplify the calculation detailed in the table by recognizing
that the salary is changing by each year. Thus, each year the new salary is of the
previous year’s salary. In other words, the yearly percentage increase is 4.8%, which
can be applied to give the new salary by multiplying by 1.048 as demonstrated
above.
You may have calculated the salaries for each year by first finding 4.8% of the previous year’s salary, and then adding that value to the previous year’s salary. Doing it this way, you have to type two calculations into a calculator (calculate 4.8%, then record that value, then enter the addition of that value to the salary). If you did it this way, think about how to write (and do) the computation with just one calculation entry into a calculator by factoring .
What if we want to know what the athlete’s salary will look like more than one year out? We can use the same process as above without computing each intermediate step to see that the salary after five years will be We can re-write this using the associative property of multiplication as: Recall from the previous section on exponents, this is the same as
Here we have demonstrated precisely how we can use an exponential expression to more efficiently calculate (and represent such a calculation) the change in salary represented by the above table.
What we have developed above is an exponential function or model to describe this athlete’s salary. Let’s look at each value in our model and identify what each piece represents.
is the athlete’s salary for year of the contract. The athlete has a starting salary of $8.3 million, from which the yearly increase is calculated (as a percentage of the original amount).
In general, an exponential model is an equation of the form
where the growth factor and is the initial value. Moreover, if , we call the
growth rate. Whenever , we often say that the model is exhibiting exponential
growth, whereas if , we say the model exhibits exponential decay. Note that when
we have exponential decay the growth rate is negative.
What we have outlined above is an understanding of what each piece of an
exponential model in the form means in terms of a given context. We can now use
this understanding to create an exponential model for exponential scenarios without
having to go through the “step by step” process as we did in the original table for the
athlete salary problem. Some CDs (a banking investment option) are offering rates
that give about 3.10% yield per year as long as a minimum amount, such as $5000, is
invested. “3.10% yield” simply means the investment increases (earns) about 3.10%.
CDs are only given for a certain amount of time (usually a certain number of years).
When a CD “matures” (ends), you usually have the option to renew the
CD.
- (a)
- Use this scenario to fill out the table below
- (b)
- Suppose you invest the minimum amount of $5000 in order to get this 3.10% yield. Write an equation to describe how much the investment will be worth after x years. Write the units of each value and identify what each represents.
- (c)
- Fill in the first column of the table below with the values you calculated
in your pre class work. Then, enter the appropriate values into your model
from above to fill in the last column of the table.
How do these values compare? Are they exactly the same? Should they be? Explain.
- (d)
- Suppose this is a 7 year CD. How much will your investment be worth at the end of the CD?
- (e)
- Suppose you keep renewing this CD with this rate every time it “matures”
(comes to the end). About how many years will it take to double the initial
investment? Use “Guess and Check” to answer this question:
- (i)
- Find the closest whole number that gives us less than we want.
- (ii)
- Find the closest whole number that gives us more than we want.
- (iii)
- Which of these values is closer to what we want? Use whichever value is closer as your estimated value for the answer.
- (a)
- (b)
-
is the amount (in dollars) that the investment is worth after years.
5000 is the amount of the initial investment (also in dollars).
1.031 is the growth rate of the investment, specifically a 3.1% increase each year, which means we have 103.1% of our initial investment after 1 year.
is the number of years of investment to reach the value .
- (c)
-
The values produced from our function match exactly those given by calculating from the previous year. Think back to our explanation in the previous example. We are performing the same calculation, just via a more efficient method.
- (d)
- To calculate our return after 7 years, we simply plug into the formula found in part (b): We are dealing with money, so we round this to the nearest hundredth, giving a total value of $6191.28. This is an increase of $1191.28 over our initial investment.
- (e)
- Doubling our investment, means we want the total value to be $10000. The
process to answer this question is as follows:
- (i)
- Begin “guessing”. The idea is to use your calculator to plug in whole
numbers for until the function produces a value less than our goal
of $10000 and such that the next whole number will be more. From
the previous part, we know that 7 years only gives about a $2000
increase, so we should begin with a number at least twice that.
Ultimately, we find that plugging in will produce a return of
which is less than $10000, while results in more (see below). - (ii)
- As noted above, will result in slightly more than the $10000 we are
looking for:
The left-hand side is the calculator-ready expression, and the right-hand side is an approximation.
- (iii)
- Finally, we must ask ourselves which of these is
closer to the full value of investment that we want?
23 years gets us closer to the goal of $10000. Hence, we must maintain the
CD for 23 years to double our investment.
Though it may seem extraneous to write out this subtraction, it is better to develop this habit of checking and concluding your work in a clear and transparent manner. For your success, we recommend you perform such calculations for every problem.
Finally, we give an example where we are looking for a value that decreases over time.
Suppose a patient is given an initial does of 300mg of a medication which degrades by
25% each hour.
- (a)
- Write a function describing the degradation of this medication. Identify the units of each value and write a sentence explaining what each piece of the function represents in this context.
- (b)
- How much drug concentration is left after one day?
- (c)
- Approximately how long until the drug concentration is halved? Estimate this value to the nearest half hour. Use “Guess and Check” to answer this question (as in the previous example).
- (d)
- Approximately how long until the drug concentration is reduced to 5mg? Estimate this value to the nearest half hour. Use “Guess and Check” to answer this question (as in the previous example).
- (a)
- , which is equivalent to .
is the number of milligrams of drug concentration hours after the initial dose was administered.
300mg is the initial dose of the medicine.
0.75 is the rate of change of the potency of the medication, specifically, a decrease of 25% in the dose each hour.
is the number of hours since the initial does was administered.
- (b)
- We measure time since the initial dose was administered in hours, so we must plug in hours (1 day) to the equation found in part (a):
- (c)
- The initial administration had a concentration of 300mg, so we are looking
for the number of hours until mg. Since we are looking to the nearest
half-hour, we plug in half-values for as well: 1, 1.5, 2, 2.5 …
The closest value for above 150mg is 2, as .The closest value for below 150mg is 2.5, as .
We now check to see which value is closer to our target concentration of 150mg:
168.75mg - 150mg = 18.75mg
150mg - 146.14mg = 3.86mg
Thus, the 2.5 hour estimate is closer. - (d)
- We now wish to find out how long it takes to reduce the concentration to
5mg. Thus, we want to solve for . Again, we are estimating to the nearest
half-hour, so we plug in half-values for .
The closest value for that is above 5mg is 14, as .
The closest value for that is below 5mg is 14.5, as .Once again, we check to see which value is closer to our target concentration of 5mg:
5.34mg - 5mg = 0.34mg
5mg - 4.63mg = 0.37mg
0.34mg is barely closer to the target concentration of 5mg, so we use 14.5 hours as an estimate for how long it takes 300mg to degrade to 5mg.
4. Suppose there is a new virus that reportedly doubles infection cases in about 20
days. What would be this virus’ infection rate (as a percent)? Hint: Set
up an exponential function, assuming that there were initially 80 recorded
infections. After you have figured out the infection rate, think about/explain
why an initial number of infections was not needed in order to answer this
question.