Which of the following is exponential growth?

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