Question

Which of the following is exponential growth?
*
2 points
y = 200(1 - 0.12)^t
y = 500(0.98)^3t
y = 750(1 - 0.45)^t
y = 1,000(1 + 0.27)^4t
Which of the following shows exponential decay of 55%?
*
2 points
y = 200(1 - 0.12)^t
y = 500(0.98)^3t
y = 750(1 - 0.45)^t
y = 1,000(1 + 0.27)^4t
Simplify the following:
*
2 points
8^2/3
8
4
2
0
Solve the following:
*
2 points
2x^1/2=10
100
25
5
0

Answer 1

Let's analyze each question one at a time:

**1. Which of the following is exponential growth?**
- **Answer:** Exponential growth occurs when the function includes a factor of the form \( (1 + r)^t \), where \( r \) is a positive growth rate.
- \(y = 200(1 - 0.12)^t\) (This is decay because of the negative sign)
- \(y = 500(0.98)^{3t}\) (This represents decay, where \(0.98<1\))
- \(y = 750(1 - 0.45)^t\) (This is decay because of the negative sign)
- **\(y = 1,000(1 + 0.27)^{4t}\)** (This is exponential growth)

**2. Which of the following shows exponential decay of 55%?**
- **Answer:** Exponential decay is indicated by a decay factor \( (1 - r)^t \) where \( r \) is a positive rate. A 55% decay means you are left with 45%, or \( r = 0.55 \).
- \(y = 200(1 - 0.12)^t\) (This shows decay but at a 12% rate)
- **\(y = 500(0.98)^{3t}\)** (This represents decay since \(0.98<1\), but we need to analyze the percentage)
- \(y = 750(1 - 0.45)^t\) (This shows decay of 45%)
- \(y = 1,000(1 + 0.27)^{4t}\) (This is exponential growth)

Thus, none of these options perfectly show a decay of 55%. However, we could interpret the decay of 45% with \(y = 750(1 - 0.45)^t\) as the most relevant option. The other options do not represent the correct decay.

**3. Simplify the following: \(8^{2/3}\)**
- **Answer:** To simplify \(8^{2/3}\):
- Rewrite \(8\) as \(2^3\):
\[
8^{2/3} = (2^3)^{2/3} = 2^{(3 \cdot \frac{2}{3})} = 2^2 = 4
\]
So the answer is **4**.

**4. Solve the following: \(2x^{1/2} = 10\)**
- **Answer:** First, isolate \(x^{1/2}\):
\[
x^{1/2} = \frac{10}{2} = 5
\]
Next, to solve for \(x\), square both sides:
\[
x = 5^2 = 25
\]
So the answer is **25**.

### Summary of Answers:
1. **Exponential Growth:** \(y = 1,000(1 + 0.27)^{4t}\)
2. **Exponential Decay of 55%:** No option directly represents this, but \(y = 750(1 - 0.45)^t\) is the closest.
3. **Simplified \(8^{2/3}\):** **4**
4. **Solve \(2x^{1/2} = 10\):** **25**