Exponential Growth and Decay worksheet with problems and solutions.

Let’s solve each part of the worksheet step by step. We’ll go one question at a time, check our work, and make sure everything is correct before writing the final answer.

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Part 1: Given the probability percent, determine the multiplier.

We convert percentages to decimals by dividing by 100.

a) 4% → 4 ÷ 100 = 0.04
b) 36% → 36 ÷ 100 = 0.36
c) 125.8% → 125.8 ÷ 100 = 1.258
d) 0.2% → 0.2 ÷ 100 = 0.002

✔ All correct.

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Part 2: State whether each equation represents growth or decay.

Remember:
- If the base (the number in parentheses after x) is greater than 1, it’s growth.
- If the base is between 0 and 1, it’s decay.

a) f(x) = 3^x → base = 3 → growth
b) f(x) = (1/2)^x → base = 0.5 → decay
c) f(x) = 1.4^x → base = 1.4 → growth
d) f(x) = 0.99^x → base = 0.99 → decay
e) f(x) = e^x → e ≈ 2.718 > 1 → growth
f) f(x) = (π/4)^x → π ≈ 3.14, so π/4 ≈ 0.785 < 1 → decay

✔ All correct.

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Part 3: Identify the necessary information and solve.

These are exponential growth/decay problems using the formula:

> A = P(1 + r)^t for growth
> A = P(1 - r)^t for decay

Where:
- A = final amount
- P = initial amount
- r = rate as decimal
- t = time

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Problem 3a:
Initial population = 1000
Growth rate = 11% per year → r = 0.11
Time = 5 years

A = 1000 × (1 + 0.11)^5
= 1000 × (1.11)^5
Calculate 1.11^5:

1.11² = 1.2321
1.11³ = 1.2321 × 1.11 ≈ 1.367631
1.11⁴ ≈ 1.367631 × 1.11 ≈ 1.51807041
1.11⁵ ≈ 1.51807041 × 1.11 ≈ 1.6850581551

So A ≈ 1000 × 1.685058 ≈ 1685.06

But the worksheet says “≈ 1685” — that’s fine, rounded to whole number.

Wait — the worksheet shows “≈ 1685” but then writes “≈ 1685.06”? Actually, looking again — the student wrote “≈ 1685” and then below it “≈ 1685.06”. Probably just showing both. But we’ll use exact calculation.

Actually, let me recalculate with calculator precision:

1.11^5 = 1.6850581551 → ×1000 = 1685.058... → rounds to 1685.06 if keeping two decimals.

But since population is usually whole numbers, maybe round to nearest whole number? The problem doesn’t specify. But the worksheet has “≈ 1685”, so perhaps they want integer.

Wait — actually, looking at the image, the student wrote:

“≈ 1685” and then below it “≈ 1685.06” — probably a mistake. Let’s stick with precise value.

But for consistency, let’s follow what’s expected. In many textbooks, they keep one or two decimals unless specified.

Actually, rechecking: the problem says “population”, which should be whole number. So likely 1685.

But let’s see what the worksheet expects — it shows “≈ 1685” and then “≈ 1685.06” — confusing. I think the intended answer is 1685.

Wait — no, in the image, under 3a, it says:

“≈ 1685” and then below it “≈ 1685.06” — probably the student did both. But mathematically, 1000*(1.11)^5 = 1685.058... so if rounding to nearest hundredth, it’s 1685.06. If to nearest whole number, 1685.

Since the problem doesn’t specify, but in context of population, often whole number. However, the worksheet seems to accept 1685.06? Wait, looking again — in the image, for 3a, it says:

“≈ 1685” and then below it “≈ 1685.06” — I think it's a typo or double-writing. Let’s calculate exactly.

Actually, let’s do it properly:

1.11^5 = 1.6850581551 → ×1000 = 1685.0581551 → so approximately 1685.06 if to two decimals.

But in the answer key shown in the image, it says “≈ 1685” — wait, no, in the red text, it says “≈ 1685” and then below “≈ 1685.06” — I think the red is the correct answer, and black is student work? Not sure.

To avoid confusion, let’s compute accurately and report as needed.

For now, I’ll use 1685.06 as the precise value, but note that sometimes it’s rounded.

Wait — looking at other parts, like 3b, they have “≈ 107.70”, so they are using two decimals.

So for 3a: 1685.06

But let’s confirm calculation:

1.11^5:

Step-by-step:

Year 0: 1000
Year 1: 1000 * 1.11 = 1110
Year 2: 1110 * 1.11 = 1232.1
Year 3: 1232.1 * 1.11 = 1367.631
Year 4: 1367.631 * 1.11 = 1518.07041
Year 5: 1518.07041 * 1.11 = 1685.0581551 → yes, 1685.06 when rounded to two decimals.

So answer is 1685.06

But in the worksheet, it might be written as 1685 — but based on calculation, we'll go with 1685.06.

Actually, in the image, for 3a, the red text says “≈ 1685” — wait, no, let me read carefully.

In the user's image description, for 3a, it says:

"a. Initial population = 1000
Growth rate = 11% per year
Time = 5 years
≈ 1685
≈ 1685.06"

And then in red, it might be the answer — but since I can't see colors well, I'll assume the calculation is what matters.

To be safe, I'll use the exact value: 1685.06

But let's move on and come back.

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Problem 3b:
Initial value = $500
Decay rate = 15% per month → r = 0.15
Time = 3 months

A = 500 × (1 - 0.15)^3 = 500 × (0.85)^3

0.85^2 = 0.7225
0.85^3 = 0.7225 × 0.85 = 0.614125

A = 500 × 0.614125 = 307.0625 → $307.06

Worksheet shows “≈ 307.06” — good.

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Problem 3c:
Initial amount = 200g
Half-life = 10 days → this means every 10 days, it halves.
Time = 30 days → number of half-lives = 30 / 10 = 3

So A = 200 × (1/2)^3 = 200 × 1/8 = 25g

Worksheet shows “25g” — correct.

Note: For half-life, we use A = P × (1/2)^(t/h), where h is half-life.

Here t=30, h=10, so exponent=3.

Yes.

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Problem 3d:
Initial amount = $1000
Interest rate = 5% compounded annually → r = 0.05
Time = 10 years

A = 1000 × (1 + 0.05)^10 = 1000 × (1.05)^10

Calculate 1.05^10:

1.05^2 = 1.1025
1.05^4 = (1.1025)^2 = 1.21550625
1.05^8 = (1.21550625)^2 ≈ 1.4774554437890625
Then 1.05^10 = 1.05^8 × 1.05^2 ≈ 1.4774554437890625 × 1.1025 ≈ ?

Compute:

1.4774554437890625 × 1.1025

First, 1.4774554437890625 × 1 = 1.4774554437890625
1.4774554437890625 × 0.1 = 0.14774554437890625
1.4774554437890625 × 0.0025 = 0.00369363860947265625

Better: 1.4774554437890625 × 1.1025 = 1.4774554437890625 × (1 + 0.1 + 0.0025) = but easier to multiply directly.

Use known value: 1.05^10 ≈ 1.6288946267774414

So A = 1000 × 1.6288946267774414 ≈ 1628.89

Worksheet shows “≈ 1628.89” — correct.

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Now, back to 3a — we have 1685.06, but in some contexts, population is integer. However, since other answers have decimals, we'll keep it as 1685.06.

But let's check the worksheet again — in the user's message, for 3a, it says "≈ 1685" and then "≈ 1685.06" — probably the first is approximate, second is precise. But in answer, we should put the calculated value.

Perhaps the worksheet expects 1685, but mathematically it's 1685.06.

To match the format, since 3b has 307.06, 3d has 1628.89, so 3a should be 1685.06.

Yes.

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Part 4: Compound interest problems.

Formula for compound interest:

> A = P(1 + r/n)^(nt)

Where:
- A = final amount
- P = principal
- r = annual interest rate (decimal)
- n = number of times compounded per year
- t = time in years

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Problem 4a:
P = $1000
r = 5% = 0.05
n = 1 (compounded annually)
t = 10 years

A = 1000 × (1 + 0.05/1)^(1×10) = 1000 × (1.05)^10 = same as 3d → $1628.89

Worksheet shows “$1628.89” — correct.

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Problem 4b:
Same P, r, t, but n = 4 (quarterly)

A = 1000 × (1 + 0.05/4)^(4×10) = 1000 × (1 + 0.0125)^40 = 1000 × (1.0125)^40

Calculate 1.0125^40:

We can compute step by step or use approximation.

1.0125^40 = ?

Note: (1.0125)^40 = e^(40 * ln(1.0125)) but better to calculate.

Known value: 1.0125^40 ≈ 1.643619463487008

So A ≈ 1000 × 1.643619463487008 ≈ 1643.62

Worksheet shows “$1643.62” — correct.

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Problem 4c:
n = 12 (monthly)

A = 1000 × (1 + 0.05/12)^(12×10) = 1000 × (1 + 0.0041667)^120

0.05/12 ≈ 0.0041666667

(1.0041666667)^120

Approximately: 1.0041666667^120 ≈ 1.647009497690282

A ≈ 1000 × 1.647009497690282 ≈ 1647.01

Worksheet shows “$1647.01” — correct.

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Problem 4d:
n = 365 (daily)

A = 1000 × (1 + 0.05/365)^(365×10) = 1000 × (1 + 0.0001369863)^3650

0.05/365 ≈ 0.0001369863

(1.0001369863)^3650 ≈ ?

This is close to continuous compounding, but let's compute.

Known value: approximately 1.648658625

A ≈ 1000 × 1.648658625 ≈ 1648.66

Worksheet shows “$1648.66” — correct.

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Now, all parts are solved.

Let me summarize the answers as required.

For Part 1: multipliers are 0.04, 0.36, 1.258, 0.002

Part 2: growth, decay, growth, decay, growth, decay

Part 3:
a) 1685.06
b) 307.06
c) 25
d) 1628.89

Part 4:
a) 1628.89
b) 1643.62
c) 1647.01
d) 1648.66

In the worksheet, for Part 3a, it might be listed as 1685, but based on calculation, it's 1685.06. Since the problem didn't specify rounding, and others have decimals, we'll use 1685.06.

But let's double-check 3a calculation once more.

P = 1000, r = 0.11, t = 5

A = 1000 * (1.11)^5

1.11^2 = 1.2321

1.11^3 = 1.2321 * 1.11 = let's compute: 1.2321 * 1.11 = 1.2321*1 + 1.2321*0.11 = 1.2321 + 0.135531 = 1.367631

1.11^4 = 1.367631 * 1.11 = 1.367631*1 + 1.367631*0.11 = 1.367631 + 0.15043941 = 1.51807041

1.11^5 = 1.51807041 * 1.11 = 1.51807041*1 + 1.51807041*0.11 = 1.51807041 + 0.1669877451 = 1.6850581551

Times 1000 = 1685.0581551 → rounded to two decimals: 1685.06

Yes.

Similarly, all others are correct.

Now, for the final answer, since the worksheet has multiple parts, I need to provide all answers.

The user said "solve the problem", and the image contains several questions, so I should list all answers.

In the response format, it says "Final Answer: [Provide only the final correct answer]", but since there are many, I need to present them clearly.

Probably, I should list them as per the worksheet sections.

Since the instruction is to provide the final answer after reasoning, and it's a multi-part problem, I'll compile all answers.

Let me write them in order.

──────────────────────────────────────

Final Answer:
Part 1:
a) 0.04
b) 0.36
c) 1.258
d) 0.002

Part 2:
a) growth
b) decay
c) growth
d) decay
e) growth
f) decay

Part 3:
a) 1685.06
b) 307.06
c) 25
d) 1628.89

Part 4:
a) 1628.89
b) 1643.62
c) 1647.01
d) 1648.66