B. Exponential Functions · Exponential Functions

Solving Problems with Exponential Decay: Half-Life, Bouncing Balls, Depreciation and Cooling

B2.3 B2.2About 75 minutes

Learning goals

I can solve half-life problems with A = A₀(0.5)^(t/h).
I can solve depreciation and bouncing-ball problems with y = a(1 − r)ᵗ and y = a(b)ⁿ.
I can solve cooling problems with T = Tᵣ + (T₀ − Tᵣ)(b)ᵗ, where the asymptote is the room temperature.
I can find an unknown time by using powers, reading a graph or systematic trial, and interpret the answer in context.

Words to know

half-life
The time for half of a radioactive substance (or any decaying amount) to remain.

depreciation
A loss in value over time, often modelled as a constant percent per year.

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

ambient (room) temperature
The temperature of the surroundings. A cooling object approaches it.

A₀
Read "A nought": the amount at time 0 (the initial amount).

Things that shrink by a fraction

Hospitals use radioactive substances, such as iodine-131, to diagnose and treat some illnesses; the half-life tells staff how long a dose stays active. Insurance companies use depreciation to value cars. Sports engineers test balls by how high they rebound. Food-safety rules care about how fast hot food cools.

All of these are exponential decay: each step keeps the same fraction of what was there. In this lesson you will solve problems for each one, using the same three strategies as for growth: substitute, read the graph, and systematic trial.

Half-life

If a substance has half-life h and starts with amount A₀, then after t time units:

A = A₀(0.5)^(t/h)

The exponent t/h counts the number of half-lives. Iodine-131 has a half-life of about 8 days. If a sample's activity is 120 units, then after 8 days 60 units remain, after 16 days 30, after 24 days 15, and so on.

The decay of a sample with a half-life of 8 days. Every 8 days the activity halves: 120, 60, 30, 15, 7.5. The curve approaches 0 but never reaches it in the model.

Worked example

Example 1: Half-life

A sample of iodine-131 has activity 120 units. Its half-life is about 8 days. (a) Find the activity after 30 days. (b) After how many whole half-lives is it below 1 unit? How many days is that?

(a) A = 120(0.5)^(30/8) = 120(0.5)^3.75 ≈ 8.9 units. (Check: 30 days is between 3 half-lives (15 units) and 4 (7.5 units). ✓)
(b) Keep halving: 120, 60, 30, 15, 7.5, 3.75, 1.875, 0.9375. After 7 half-lives it is below 1 unit.
7 × 8 = 56 days.

(a) about 8.9 units (b) 7 half-lives, about 56 days

Worked example

Example 2: Depreciation

A new car costs $32 000 and loses 15% of its value each year. (a) Write a model. (b) What is it worth after 5 years? (c) When is it first worth less than half the price?

(a) Decay factor 1 − 0.15 = 0.85: V = 32 000(0.85)ᵗ.
(b) V(5) = 32 000(0.85)⁵ ≈ $14 198.57.
(c) Half the price is $16 000. Trial: V(4) ≈ $16 704.20 (still above); V(5) ≈ $14 198.57 (below). So it falls below half its price during the 5th year.
Note: losing 15% a year for 5 years is not a 75% loss. The car keeps about 44% of its value, because each 15% is taken from a smaller amount.

V = 32 000(0.85)ᵗ; $14 198.57; during the 5th year

The car's value each year (points) on the model curve. The value drops below the $16 000 line between t = 4 and t = 5.

Worked example

Example 3: Bouncing ball

A ball is dropped from 2 m and each rebound reaches 65% of the previous height. (a) How high is the 3rd rebound? (b) On which bounce is the rebound first below 10 cm?

Model: h = 2(0.65)ⁿ, with h in metres and n the bounce number (whole numbers only).
(a) h(3) = 2(0.65)³ ≈ 0.549 m, about 55 cm.
(b) 10 cm = 0.1 m. Trial: h(6) ≈ 0.151 m; h(7) ≈ 0.098 m. The 7th rebound is the first below 10 cm.

(a) about 55 cm (b) the 7th bounce

Worked example

Example 4: Cooling

Soup at 90°C is set out in a 20°C kitchen. A model is T = 20 + 70(0.9)ᵗ, t in minutes. (a) Find the temperature after 5 minutes. (b) About when has it cooled to 50°C?

The 70 is the starting difference from room temperature (90 − 20); only the difference decays.
(a) T(5) = 20 + 70(0.9)⁵ ≈ 20 + 41.3 = 61.3°C.
(b) We need 70(0.9)ᵗ = 30. Trial: t = 8 gives T ≈ 50.1°C; t = 9 gives T ≈ 47.1°C. So it reaches 50°C just after 8 minutes.

(a) about 61.3°C (b) just after 8 minutes

Interpreting answers in context

Give units (days, $, cm, °C) and round sensibly: money to the cent, heights to the cm, temperatures to 0.1°C.
Say "during the 5th year" or "just after 8 minutes" when the answer falls between two trials.
Check that the answer makes sense: decay values must stay positive; a cooling object stays above room temperature; a ball cannot bounce 3.5 times.
Remember that every model is an approximation. A real ball eventually stops; real depreciation depends on the car.

Just for fun

Why don't radioactive atoms ever finish their homework?

Every time they get halfway, they take a break, then do only half of what is left. (Mathematically, a decay model never quite reaches zero!)

Hands-on activity

Half-life with a deck of cards

You need: a standard deck of 52 playing cards, a coin, paper and a pencil.

Shuffle the deck. Each card is an "atom". Record round 0: 52 atoms.
Deal the cards into two piles, one card at a time, alternating piles.
Toss the coin to choose one pile. Those atoms have "decayed": set that pile aside. Record how many cards are left.
Shuffle the cards that are left and repeat steps 2 and 3 until only one card remains.
Make a table of round number n and cards left, graph it, and compare with the model N = 52(0.5)ⁿ.

What to look for: the counts are 52, 26, 13, then 6 or 7 (an odd pile splits unevenly), then about 3, 2 and 1. They follow the model N = 52(0.5)ⁿ closely, and each round plays the role of one half-life. The graph is a decreasing exponential curve that flattens out near 0.

Your turn

A substance has half-life 5 years and starts at 80 g. How much remains after 15 years?

Same substance: how much remains after 12 years (one decimal place)?

A $1200 phone loses 25% of its value each year. Write a model and find its value after 3 years.

A ball dropped from 150 cm rebounds to 80% of its previous height. How high is the 4th rebound?

T = 21 + 60(0.85)ᵗ (°C, minutes). Find T after 10 minutes.

A sample of 64 mg has half-life 3 hours. When are 4 mg left?

Explain why the depreciation model V = 32 000(0.85)ᵗ never gives $0, but a real car might be worth $0.

More practice: warm-up

What fraction remains after 2 half-lives?

What decay factor matches a 20% loss per year?

Evaluate 500(0.5)³.

A ball rebounds to 1/2 of its height. Dropped from 4 m, how high is the 2nd rebound?

In T = 20 + 70(0.9)ᵗ, what temperature does the object approach?

More practice: on your own

Carbon-14 has a half-life of about 5730 years. What percent of the original carbon-14 remains after 11 460 years?

A snowmobile bought for $14 000 depreciates 12% a year. In which year does it first fall below $8000?

A ball dropped from 3 m rebounds to 70% each time. On which bounce is the rebound first under 50 cm?

Tea at 95°C cools in a 22°C room: T = 22 + 73(0.92)ᵗ. Use systematic trial to find when it reaches 60°C (nearest minute).

A pond's algae coverage shrinks by 30% each week after treatment, from 400 m². Write the model and find the area after 6 weeks.

Challenge

A radioactive sample drops from 200 units to 25 units in 18 hours. Find its half-life.

A car loses 15% of its value each year. A student says, "After 3 years it has lost 45%." Find the true percent lost.

Two balls are dropped from 2 m. Ball A rebounds to 75% each time; ball B to 60%. After how many bounces is ball A's rebound more than 3 times ball B's?

Check your understanding

Quiz

A substance has a half-life of 10 days. What fraction remains after 30 days?

A $20 000 vehicle depreciates 10% per year. Which expression gives its value after 6 years?

A ball dropped from 5 m rebounds to 60% each time. How high is the 2nd rebound?

For T = 18 + 62(0.88)ᵗ, which statement is true?

An amount decays from 400 to 50 in 12 hours. What is the half-life?

A car's value is $9800 after 6 years and $8330 after 7 years. Which conclusion is correct about when it is worth $9000?

For parents and teachers

Big idea (B2.3, with B2.2): students solve real decay problems (radioactive half-life, depreciation, bouncing balls, cooling) by substituting into equations and interpreting graphs, and they state answers sensibly in context.

How to help: Ask your teen to estimate before calculating ("30 days is a bit less than 4 half-lives, so a bit more than 1/16 is left"). The deck-of-cards activity takes 10 minutes and makes half-life concrete.

Common difficulties:

Using r instead of 1 − r: 15% depreciation is a factor of 0.85.
Adding percents: three years of 15% loss is about 38.6%, not 45%.
Cooling models: only the difference from room temperature decays: T = 20 + 70(0.9)ᵗ, not 90(0.9)ᵗ.
Bounce numbers: answers must be whole bounces.

Facts used: the half-lives of iodine-131 (about 8 days) and carbon-14 (about 5730 years) are standard published values; other numbers are realistic examples.