Exponential Growth and Decay Worksheet - Free Printable

Problem Analysis:

The task involves solving exponential growth and decay problems using the general formula for exponential functions:

1. Exponential Growth Formula:
$$
y = a \cdot (1 + r)^t
$$
where:
- $ y $ is the value after time $ t $,
- $ a $ is the initial value,
- $ r $ is the growth rate (expressed as a decimal),
- $ t $ is the time in years.

2. Exponential Decay Formula:
$$
y = a \cdot (1 - r)^t
$$
where:
- $ y $ is the value after time $ t $,
- $ a $ is the initial value,
- $ r $ is the decay rate (expressed as a decimal),
- $ t $ is the time in years.

Let's solve each problem step by step.

---

Problem 1: McDonald's Sales

- Initial sales ($a$): $650,000
- Growth rate ($r$): 4% per year (or $0.04$)
- Time ($t$): 3 years

Using the exponential growth formula:
$$
y = a \cdot (1 + r)^t
$$
Substitute the values:
$$
y = 650,000 \cdot (1 + 0.04)^3
$$
Calculate step by step:
1. Compute $ 1 + 0.04 $:
$$
1 + 0.04 = 1.04
$$
2. Raise $ 1.04 $ to the power of 3:
$$
1.04^3 = 1.04 \times 1.04 \times 1.04 = 1.124864
$$
3. Multiply by the initial value:
$$
y = 650,000 \cdot 1.124864 = 731,161.6
$$

Thus, the sales after 3 years are:
$$
\boxed{731,161.6}
$$

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Problem 2: School Population

- Initial population ($a$): 800 students
- Growth rate ($r$): 3% per year (or $0.03$)
- Time ($t$): 6 years

Using the exponential growth formula:
$$
y = a \cdot (1 + r)^t
$$
Substitute the values:
$$
y = 800 \cdot (1 + 0.03)^6
$$
Calculate step by step:
1. Compute $ 1 + 0.03 $:
$$
1 + 0.03 = 1.03
$$
2. Raise $ 1.03 $ to the power of 6:
$$
1.03^6 = 1.03 \times 1.03 \times 1.03 \times 1.03 \times 1.03 \times 1.03 \approx 1.194052
$$
3. Multiply by the initial value:
$$
y = 800 \cdot 1.194052 = 955.2416
$$

Since population must be a whole number, round to the nearest whole number:
$$
y \approx 955
$$

Thus, the population after 6 years is:
$$
\boxed{955}
$$

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Problem 3: Bacteria Population

- Initial population ($a$): 20 bacteria
- Growth rate ($r$): 10% per hour (or $0.10$)
- Time ($t$): 10 hours

Using the exponential growth formula:
$$
y = a \cdot (1 + r)^t
$$
Substitute the values:
$$
y = 20 \cdot (1 + 0.10)^{10}
$$
Calculate step by step:
1. Compute $ 1 + 0.10 $:
$$
1 + 0.10 = 1.10
$$
2. Raise $ 1.10 $ to the power of 10:
$$
1.10^{10} = 1.10 \times 1.10 \times \cdots \times 1.10 \approx 2.593742
$$
3. Multiply by the initial value:
$$
y = 20 \cdot 2.593742 = 51.87484
$$

Since the population must be a whole number, round to the nearest whole number:
$$
y \approx 52
$$

Thus, the population after 10 hours is:
$$
\boxed{52}
$$

---

Problem 4: Furniture Store Orders

- Initial orders ($a$): $231,200
- Growth rate ($r$): 6.7% per year (or $0.067$)
- Time ($t$): 5 years

Using the exponential growth formula:
$$
y = a \cdot (1 + r)^t
$$
Substitute the values:
$$
y = 231,200 \cdot (1 + 0.067)^5
$$
Calculate step by step:
1. Compute $ 1 + 0.067 $:
$$
1 + 0.067 = 1.067
$$
2. Raise $ 1.067 $ to the power of 5:
$$
1.067^5 = 1.067 \times 1.067 \times \cdots \times 1.067 \approx 1.391089
$$
3. Multiply by the initial value:
$$
y = 231,200 \cdot 1.391089 = 321,999.98
$$

Thus, the orders after 5 years are:
$$
\boxed{321,999.98}
$$

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Problem 5: India's Population

- Initial population ($a$): 273,000,000 (in 1999)
- Growth rate ($r$): 2.0% per year (or $0.02$)
- Time ($t$): 8 years (from 1999 to 2007)

Using the exponential growth formula:
$$
y = a \cdot (1 + r)^t
$$
Substitute the values:
$$
y = 273,000,000 \cdot (1 + 0.02)^8
$$
Calculate step by step:
1. Compute $ 1 + 0.02 $:
$$
1 + 0.02 = 1.02
$$
2. Raise $ 1.02 $ to the power of 8:
$$
1.02^8 = 1.02 \times 1.02 \times \cdots \times 1.02 \approx 1.171659
$$
3. Multiply by the initial value:
$$
y = 273,000,000 \cdot 1.171659 = 319,599,967
$$

Thus, the population in 2007 is:
$$
\boxed{319,599,967}
$$

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Final Answers:

1. $ \boxed{731,161.6} $
2. $ \boxed{955} $
3. $ \boxed{52} $
4. $ \boxed{321,999.98} $
5. $ \boxed{319,599,967} $