B. Exponential Functions · Exponential Functions

Solving Problems with Exponential Models

B3.3About 75 minutes

Learning goals

I can solve problems about radioactive decay, population growth, bouncing balls and compound interest.
I can find an output by substituting into an exponential equation or reading a graph.
I can find when a model reaches a target value using a graph, systematic trial or a common base.
I can communicate answers with units, sensible rounding and a reasonableness check.

Words to know

substitute
Replace the variable with a number to calculate the output, for example t = 24 in A = 200(1/2)^(t/8).

guess and check (systematic trial)
Trying values of the input, getting closer each time, to find when a model reaches a target value.

compound interest
Interest paid on the principal and on interest already earned: A = P(1 + i)ⁿ.

compounding period
How often interest is added: annually, semi-annually, monthly, and so on. i is the rate per period and n the number of periods.

rebound ratio
The fraction of its previous height that a ball reaches on each bounce.

Putting the models to work

This lesson pulls the unit together. You will answer real questions using an exponential graph or equation: How much is left? How long until…? How high after…? Two kinds of questions come up:

Find the output for a given input: substitute into the equation, or read the graph.
Find the input that gives a target output: read the graph, use systematic trial (guess and check), or, when possible, write both sides with a common base (Lesson 10). In MHF4U you will learn logarithms, which do this exactly.

A problem-solving routine

Identify a (initial amount), b (or the half-life / doubling time) and what each variable means, with units.
Write the model if it is not given.
Decide: find the output (substitute) or find the input (graph or trial).
Calculate, keeping full calculator values until the end.
Answer in a sentence with units and sensible rounding, and check that it is reasonable (decay should go down, growth up).

Worked example

Example 1: Radioactive decay

Iodine-131, used in some medical treatments, has a half-life of about 8 days. A sample contains 200 mg. (a) Write a model. (b) How much remains after 24 days? After 30 days? (c) About when will less than 10 mg remain?

(a) A = 200(1/2)^(t/8), with A in milligrams and t in days.
(b) t = 24: 24/8 = 3 half-lives, so A = 200(1/8) = 25 mg. t = 30: A = 200(0.5)^3.75 ≈ 14.9 mg.
(c) Trial: t = 32 gives 200(1/2)⁴ = 12.5 mg (too much). t = 40 gives 6.25 mg (below 10). Narrow down: t = 34 gives ≈ 10.5 mg; t = 35 gives ≈ 9.6 mg.
So the amount falls below 10 mg at about t = 35 days (more precisely, between day 34 and day 35).

25 mg after 24 days; about 14.9 mg after 30 days; below 10 mg after about 35 days.

Every 8 days the amount halves: 200 → 100 → 50 → 25 → 12.5 mg. The curve crosses the 10 mg line between day 32 and day 40, at about day 35.

Worked example

Example 2: Population growth

A town of 12 000 people is growing at 2.5% per year. (a) Write a model. (b) Predict the population in 10 years. (c) About how long will it take to double?

(a) P = 12 000(1.025)ᵗ, t in years.
(b) P(10) = 12 000(1.025)¹⁰ ≈ 12 000(1.2801) ≈ 15 361 people.
(c) Doubling means P = 24 000, so we need (1.025)ᵗ = 2. Trial: 1.025²⁰ ≈ 1.64; 1.025²⁸ ≈ 1.996; 1.025²⁹ ≈ 2.046.
So it doubles in about 28 years (just over 28).

About 15 361 people in 10 years; doubles in about 28 years.

Worked example

Example 3: A bouncing ball

A ball is dropped from 3 m. On each bounce it rises to 65% of its previous height. (a) Write a model for the height of the nth rebound. (b) Find the height of the 4th rebound. (c) On which rebound is the height first less than 10 cm?

(a) h = 3(0.65)ⁿ, h in metres, n = 1, 2, 3, … (a discrete model).
(b) h = 3(0.65)⁴ ≈ 3(0.1785) ≈ 0.54 m (about 54 cm).
(c) 10 cm = 0.1 m. n = 7: 3(0.65)⁷ ≈ 0.147 m. n = 8: 3(0.65)⁸ ≈ 0.096 m.
The 8th rebound is the first under 10 cm.

h = 3(0.65)ⁿ; about 0.54 m on the 4th rebound; the 8th rebound is the first under 10 cm.

Worked example

Example 4: Compound interest

Maya invests $2000 at 4% per year, compounded annually. (a) Find the amount after 5 years. (b) How would the amount change at 4.8% per year compounded monthly for 5 years?

(a) A = P(1 + i)ⁿ with P = 2000, i = 0.04, n = 5: A = 2000(1.04)⁵ ≈ $2433.31.
(b) Monthly compounding: i = 0.048 ÷ 12 = 0.004 per month and n = 5 × 12 = 60 months.
A = 2000(1.004)⁶⁰ ≈ $2541.28.
The investment is the exponential function A = 2000(1.04)ᵗ evaluated at whole numbers of years. The Financial Applications unit of this course develops compound interest much further.

$2433.31; $2541.28.

Worked example

Example 5: Reading a graph

Carbon-14 has a half-life of about 5730 years. Using a graph of the fraction remaining, F = (1/2)^(t/5730), estimate what fraction of the original carbon-14 remains after 11 460 years and after 20 000 years.

11 460 = 2 × 5730: two half-lives, so F = (1/2)² = 1/4. On the graph, this is the point (11 460, 0.25).
20 000 years is between 3 half-lives (17 190 years, F = 1/8 = 0.125) and 4 half-lives (22 920 years, F = 1/16 ≈ 0.063). The graph gives about 0.09.
Substituting checks it: (1/2)^(20000/5730) ≈ (1/2)^3.49 ≈ 0.089, about 9%.

1/4 after 11 460 years; about 9% after 20 000 years.

Three ways to answer exponential questions. Substitution finds outputs; graphs, trial or a common base find inputs.

Just for fun

Why do radioactive cats have eighteen half-lives?

Because nine lives, divided in half, is still a lot of lives.

Hands-on activity

Bouncing-ball investigation

You need: a ball that bounces well (a tennis ball, rubber ball or basketball), a metre stick or tape measure, painter's tape or sticky notes, a partner, and a phone camera if you have one (slow-motion video helps).

Tape the metre stick (or tape measure) upright against a wall on a hard floor.
Drop the ball from 100 cm (do not throw it). Your partner watches the bottom of the ball and marks the highest point it reaches on each rebound. Filming and pausing makes this much easier.
Record the heights of at least 4 rebounds. Repeat the drop 3 times and average each rebound.
Compute the rebound ratios and their mean, and write the model h = 100(b)ⁿ.
Use your model to predict the 6th rebound and the first rebound below 5 cm. Test the prediction if you can.
Try a different ball or a different floor (carpet vs. tile). How does b change?

What to look for: many balls give ratios somewhere between 0.5 and 0.8, and the ratio stays roughly constant, so the heights follow an exponential decay model. Carpet absorbs more energy, so b is smaller. Predictions for later bounces are less accurate because the heights are small and hard to measure.

Your turn

A substance has a half-life of 10 years. How much of a 60 g sample remains after 30 years?

A population P = 5000(1.03)ᵗ. Find P after 8 years.

$1500 is invested at 5% per year compounded annually. Find the amount after 6 years.

A ball dropped from 2 m rebounds to 70% of its height each bounce. Find the height of the 3rd rebound.

For the ball above, on which rebound is the height first below 0.5 m?

A culture of 800 bacteria doubles every 2 hours. About how many are there after 7 hours?

Using systematic trial, find the whole number of years for $1000 at 6% compounded annually to first exceed $1500.

More practice: warm-up

How much of 80 mg remains after 2 half-lives?

Write 6% growth per year as a multiplier.

What is 500(1/2)³?

A ball rebounds to 50% each bounce from 4 m. Height after 2 bounces?

In A = P(1 + i)ⁿ, what does n count?

More practice: on your own

A car worth $28 000 loses 15% of its value each year. When does its value first fall below $10 000? (Whole years.)

Find the amount of $3000 at 6% per year compounded quarterly for 4 years.

A medicine leaves the body so that 75% remains after each hour. Starting with 400 mg, how much remains after 5 hours? Write the model.

A rumour starts with 3 people and the number who have heard it triples every day. Using a common base, find when 243 people will have heard it.

A radioactive sample drops from 640 mg to 40 mg in 20 days. Find the half-life.

Challenge

Two investments: A: $5000 at 4% compounded annually. B: $4500 at 5% compounded annually. After how many whole years is B worth more than A?

A ball is dropped from 4 m and rebounds to 60% of each height. Find the total distance it has travelled when it hits the floor for the 3rd time.

A town grows by 3% a year while a neighbouring town shrinks by 3% a year. Both start at 20 000. After 10 years, is the difference between them 2 × 3% × 10 × 20 000 = 12 000? Explain.

Check your understanding

Quiz

A 160 mg sample has a half-life of 5 hours. How much remains after 15 hours?

What is the value of $1000 invested at 3% compounded annually for 2 years?

A ball dropped from 5 m rebounds to 40% of its height. Height of the 2nd rebound?

P = 8000(1.02)ᵗ. Which is closest to P after 15 years?

Using a common base, when does 5(2)ᵗ reach 160?

Which method finds when a model reaches a target value, without logarithms?

Which model fits a car that loses 20% of its value each year, starting at $25 000?

For parents and teachers

Big idea: Expectation B3.3: solve problems using given graphs or equations of exponential functions from real-world applications (radioactive decay, population growth, height of a bouncing ball, compound interest) by interpreting graphs or substituting values for the exponent.

How to help: Ask your teen to estimate before calculating ("should the answer be bigger or smaller than we started with?"). For "how long until…" questions, encourage an organized trial table rather than random guesses.

Common difficulties:

Using the half-life itself as the exponent: after 24 days with an 8-day half-life, the exponent is 24/8 = 3.
Compound interest: forgetting to divide the annual rate by the number of periods per year (to get i) and to multiply the years by it (to get n).
Rounding too early, which can change the answer to a "first time below" question.
Answering with a bare number. Every answer needs units and a sentence.