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
Captionless Image
8
4
2
0
Solve the following:
*
2 points
Captionless Image
100
25
5
0

Answer 1

To identify exponential growth and decay and simplify the expressions, let's analyze each option.

### Exponential Growth
Exponential growth takes the form \(y = a(1 + r)^t\), where \(r\) is a positive rate.

1. \(y = 200(1 - 0.12)^t\) - This is not exponential growth; it's a decay scenario.
2. \(y = 500(0.98)^{3t}\) - This represents exponential decay, since \(0.98 < 1\).
3. \(y = 750(1 - 0.45)^t\) - This is also not exponential growth; it's decay.
4. \(y = 1,000(1 + 0.27)^{4t}\) - This represents exponential growth because \(1 + 0.27 = 1.27 > 1\).

Thus, the answer for exponential growth is:
**y = 1,000(1 + 0.27)^{4t}**

### Exponential Decay of 55%
To show exponential decay of 55%, we can express it as:
\(y = a(1 - r)^t\) where \(r = 0.55\), meaning \(1 - r = 0.45\).

1. \(y = 200(1 - 0.12)^t\) - Not decay of 55%.
2. \(y = 500(0.98)^{3t}\) - Not decay of 55%.
3. \(y = 750(1 - 0.45)^t\) - This indicates decay of 45%, not 55%.
4. \(y = 1,000(1 + 0.27)^{4t}\) - Not decay; it's growth.

In fact, none of these options represent decay of 55% directly, however, with the mentioned consideration of \(r = 0.55\), the ideal format would be \(y = a(1 - 0.55)^t\).
Thus, the answer is:
**None of the provided options show 55% decay.**

### Simplification
To simplify an expression, I would need the specific mathematical expression to work with. If simplifying a fraction, for instance:
\[
\frac{8}{4}
\]
This simplifies to:
\[
2
\]

### Solve the following
Without a specified equation or expression, I cannot provide an answer. If provided, the numerical calculation will be performed accordingly.

Please provide explicit mathematical expressions for simplification or solving that you would like assistance with!