Exponential Growth And Decay Worksheet - Fill and Sign Printable ... - Free Printable

Problem Analysis:

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

\[
y = a(1 + r)^t
\]

Where:
- \( y \) is the final amount after time \( t \).
- \( a \) is the initial amount.
- \( r \) is the rate of growth (or decay, if negative).
- \( t \) is the time period.

We will solve each problem step by step.

---

Problem 1: Investment Growth

Given:
- Initial value of investment (\( a \)) = $8000.
- Annual interest rate (\( r \)) = 6% = 0.06.
- Time (\( t \)) = 25 years.

Formula:
\[
y = a(1 + r)^t
\]

Substitute the values:
\[
y = 8000(1 + 0.06)^{25}
\]
\[
y = 8000(1.06)^{25}
\]

Calculate \( (1.06)^{25} \):
Using a calculator:
\[
(1.06)^{25} \approx 4.29187
\]

Final calculation:
\[
y = 8000 \times 4.29187 \approx 34334.96
\]

Answer:
\[
\boxed{34334.96}
\]

---

Problem 2: Cost of College Tuition

Given:
- Initial cost of college tuition (\( a \)) = $13,000.
- Annual increase rate (\( r \)) = 6% = 0.06.
- Time (\( t \)) = 4 years.

Formula:
\[
y = a(1 + r)^t
\]

Substitute the values:
\[
y = 13000(1 + 0.06)^4
\]
\[
y = 13000(1.06)^4
\]

Calculate \( (1.06)^4 \):
Using a calculator:
\[
(1.06)^4 \approx 1.262477
\]

Final calculation:
\[
y = 13000 \times 1.262477 \approx 16412.20
\]

Answer:
\[
\boxed{16412.20}
\]

---

Problem 3: Number of Students at a High School

Given:
- Initial number of students (\( a \)) = 300.
- Annual increase rate (\( r \)) = 8% = 0.08.
- Time (\( t \)) = 5 years.

Formula:
\[
y = a(1 + r)^t
\]

Substitute the values:
\[
y = 300(1 + 0.08)^5
\]
\[
y = 300(1.08)^5
\]

Calculate \( (1.08)^5 \):
Using a calculator:
\[
(1.08)^5 \approx 1.469328
\]

Final calculation:
\[
y = 300 \times 1.469328 \approx 440.7984
\]

Since the number of students must be a whole number, round to the nearest whole number:
\[
y \approx 441
\]

Answer:
\[
\boxed{441}
\]

---

Problem 4: Annual Sales

Given:
- Initial annual sales (\( a \)) = $149,000.
- Annual increase rate (\( r \)) = 6% = 0.06.
- Time (\( t \)) = 7 years.

Formula:
\[
y = a(1 + r)^t
\]

Substitute the values:
\[
y = 149000(1 + 0.06)^7
\]
\[
y = 149000(1.06)^7
\]

Calculate \( (1.06)^7 \):
Using a calculator:
\[
(1.06)^7 \approx 1.50363
\]

Final calculation:
\[
y = 149000 \times 1.50363 \approx 224049.87
\]

Answer:
\[
\boxed{224049.87}
\]

---

Problem 5: Population of a Small Town

Given:
- Initial population (\( a \)) = 1000.
- Annual growth rate (\( r \)) = 3% = 0.03.
- Time (\( t \)) = 10 years.

Formula:
\[
y = a(1 + r)^t
\]

Substitute the values:
\[
y = 1000(1 + 0.03)^{10}
\]
\[
y = 1000(1.03)^{10}
\]

Calculate \( (1.03)^{10} \):
Using a calculator:
\[
(1.03)^{10} \approx 1.343916
\]

Final calculation:
\[
y = 1000 \times 1.343916 \approx 1343.916
\]

Since population must be a whole number, round to the nearest whole number:
\[
y \approx 1344
\]

Answer:
\[
\boxed{1344}
\]

---

Problem 6: Number of Cell Phone Subscribers

Given:
- Initial number of subscribers (\( a \)) = 285.
- Annual increase rate (\( r \)) = 75% = 0.75.
- Time (\( t \)) = 3 years.

Formula:
\[
y = a(1 + r)^t
\]

Substitute the values:
\[
y = 285(1 + 0.75)^3
\]
\[
y = 285(1.75)^3
\]

Calculate \( (1.75)^3 \):
Using a calculator:
\[
(1.75)^3 \approx 5.359375
\]

Final calculation:
\[
y = 285 \times 5.359375 \approx 1530.9375
\]

Since the number of subscribers must be a whole number, round to the nearest whole number:
\[
y \approx 1531
\]

Answer:
\[
\boxed{1531}
\]

---

Final Answers:

1. \(\boxed{34334.96}\)
2. \(\boxed{16412.20}\)
3. \(\boxed{441}\)
4. \(\boxed{224049.87}\)
5. \(\boxed{1344}\)
6. \(\boxed{1531}\)