10 6 exponential growth and decay worksheet answer key Doc ... - Free Printable

Let's solve this step by step.

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Part 1: Basic Equation Statements

We are given the exponential formula:

$$
y = a(1 \pm r)^x
$$

Where:
- $ a $ = initial value
- $ r $ = percent of change (as a decimal)
- If $ +r $: growth
- If $ -r $: decay

We will fill in the table for each equation.

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#### 1) $ y = 7500(1 + 0.03)^x $

- Initial Value: $ a = 7500 $
- Growth or Decay? → Growth (since we add)
- Percent of Change: $ r = 0.03 = 3\% $

✔ Answer:
Initial Value: 7500
Growth or Decay: Growth
Percent of Change: 3%

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#### 2) $ y = (0.85)^x $

This can be rewritten as:
$ y = 1(0.85)^x $ → so $ a = 1 $, and $ 0.85 = 1 - 0.15 $

- Initial Value: $ a = 1 $
- Growth or Decay? → Decay (since it's less than 1)
- Percent of Change: $ r = 0.15 = 15\% $

✔ Answer:
Initial Value: 1
Growth or Decay: Decay
Percent of Change: 15%

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#### 3) $ y = 900(1.27)^x $

- $ a = 900 $
- $ 1.27 = 1 + 0.27 $ → growth
- $ r = 0.27 = 27\% $

✔ Answer:
Initial Value: 900
Growth or Decay: Growth
Percent of Change: 27%

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#### 4) $ y = 24000(1 - 0.08)^t $

- $ a = 24000 $
- $ 1 - 0.08 = 0.92 $ → decay
- $ r = 0.08 = 8\% $

✔ Answer:
Initial Value: 24000
Growth or Decay: Decay
Percent of Change: 8%

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#### 5) $ y = 750(1.15)^t $

- $ a = 750 $
- $ 1.15 = 1 + 0.15 $ → growth
- $ r = 0.15 = 15\% $

✔ Answer:
Initial Value: 750
Growth or Decay: Growth
Percent of Change: 15%

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#### 6) $ y = 250,000(0.65)^t $

- $ a = 250,000 $
- $ 0.65 = 1 - 0.35 $ → decay
- $ r = 0.35 = 35\% $

✔ Answer:
Initial Value: 250,000
Growth or Decay: Decay
Percent of Change: 35%

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#### 7) $ y = 11,275(1.1)^t $

- $ a = 11,275 $
- $ 1.1 = 1 + 0.1 $ → growth
- $ r = 0.1 = 10\% $

✔ Answer:
Initial Value: 11,275
Growth or Decay: Growth
Percent of Change: 10%

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#### 8) $ y = 673(2.2)^x $

- $ a = 673 $
- $ 2.2 = 1 + 1.2 $ → growth (very fast!)
- $ r = 1.2 = 120\% $

✔ Answer:
Initial Value: 673
Growth or Decay: Growth
Percent of Change: 120%

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✔ Completed Table:

| EQUATION | Initial Value | Growth or Decay? | Percent of Change |
|--------|----------------|--------------------|---------------------|
| 1) $ y = 7500(1 + 0.03)^x $ | 7500 | Growth | 3% |
| 2) $ y = (0.85)^x $ | 1 | Decay | 15% |
| 3) $ y = 900(1.27)^x $ | 900 | Growth | 27% |
| 4) $ y = 24000(1 - 0.08)^t $ | 24000 | Decay | 8% |
| 5) $ y = 750(1.15)^t $ | 750 | Growth | 15% |
| 6) $ y = 250,000(0.65)^t $ | 250,000 | Decay | 35% |
| 7) $ y = 11,275(1.1)^t $ | 11,275 | Growth | 10% |
| 8) $ y = 673(2.2)^x $ | 673 | Growth | 120% |

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Part 2: Word Problems

Use the formula:
$$
y = a(1 \pm r)^t
$$

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#### 1. Car Depreciation

- Initial price: $ a = 10,000 $
- Annual depreciation: 10% → $ r = 0.10 $
- Decay: use $ (1 - r) = 0.90 $
- Time: $ t = 8 $ years

$$
y = 10000(1 - 0.10)^8 = 10000(0.90)^8
$$

Calculate:

$$
0.90^8 = (0.9)^8 \approx 0.430467
$$

$$
y \approx 10000 \times 0.430467 = 4304.67
$$

✔ PRICE: $4,304.67

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#### 2. Baseball Card Appreciation

- Initial value: $ a = 5 $
- Annual appreciation: 5% → $ r = 0.05 $
- Growth: $ (1 + 0.05) = 1.05 $
- Time: $ t = 25 $

$$
y = 5(1.05)^{25}
$$

Calculate:

$$
1.05^{25} \approx 3.38635
$$

$$
y \approx 5 \times 3.38635 = 16.93175
$$

✔ VALUE: $16.93 (rounded to nearest cent)

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#### 3. Town Population Growth

- Initial population: $ a = 3200 $
- Growth rate: 25% → $ r = 0.25 $
- Growth factor: $ 1.25 $
- Time: $ t = 10 $

$$
y = 3200(1.25)^{10}
$$

Calculate:

$$
1.25^{10} \approx 9.3132
$$

$$
y \approx 3200 \times 9.3132 = 29,802.24
$$

✔ Population: 29,802 (rounded to nearest person)

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#### 4. City Pollution & Decline

- Initial population: $ a = 100,000 $
- Decrease: 2% annually → $ r = 0.02 $
- Decay factor: $ 1 - 0.02 = 0.98 $
- Time: $ t = 10 $

$$
y = 100000(0.98)^{10}
$$

Calculate:

$$
0.98^{10} \approx 0.8171
$$

$$
y \approx 100000 \times 0.8171 = 81,710
$$

✔ Population: 81,710

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✔ Final Answers:

#### Word Problems:

1. PRICE: $ \boxed{4,304.67} $
2. VALUE: $ \boxed{16.93} $
3. Population: $ \boxed{29,802} $
4. Population: $ \boxed{81,710} $

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