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Identify whether each exponential expression models growth or decay, and specify the rate.

[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{Expression} & \text{Growth or Decay} & \text{Rate} \\
\hline
30(1.25)^x & \text{Growth} & 25\% \\
500(0.75)^x & \text{Decay} & 25\% \\
2(2)^x & \text{Growth} & 100\% \\
4,000(1.01)^x & \text{Growth} & 1\% \\
7,000(0.99)^x & \text{Decay} & 1\% \\
\hline
\end{array}
\][/tex]

Sagot :

GinnyAnswer
Let's analyze each exponential expression and determine if it models growth or decay, and what the rate is. Here is a step-by-step breakdown for each expression:

1. For the expression [tex]\( 30(1.25)^x \)[/tex]:
- This is an exponential growth model because the base, 1.25, is greater than 1.
- The percentage rate of growth is calculated based on the factor 1.25. The rate is [tex]\( (1.25 - 1) \times 100\% = 0.25 \times 100\% = 25\% \)[/tex].

2. For the expression [tex]\( 500(0.75)^x \)[/tex]:
- This is an exponential decay model because the base, 0.75, is less than 1.
- The percentage rate of decay is calculated based on the factor 0.75. The rate is [tex]\( (1 - 0.75) \times 100\% = 0.25 \times 100\% = 25\% \)[/tex].

3. For the expression [tex]\( 2(2)^x \)[/tex]:
- This is an exponential growth model because the base, 2, is greater than 1.
- The percentage rate of growth is calculated based on the factor 2. The rate is [tex]\( (2 - 1) \times 100\% = 1 \times 100\% = 100\% \)[/tex].

4. For the expression [tex]\( 4,000(1.01)^x \)[/tex]:
- This is an exponential growth model because the base, 1.01, is greater than 1.
- The percentage rate of growth is calculated based on the factor 1.01. The rate is [tex]\( (1.01 - 1) \times 100\% = 0.01 \times 100\% = 1\% \)[/tex].

5. For the expression [tex]\( 7,000(0.99)^x \)[/tex]:
- This is an exponential decay model because the base, 0.99, is less than 1.
- The percentage rate of decay is calculated based on the factor 0.99. The rate is [tex]\( (1 - 0.99) \times 100\% = 0.01 \times 100\% = 1\% \)[/tex].

Now we compile this information into the table:

[tex]\[ \begin{array}{|c|c|c|} \hline \text{Expression} & \text{Growth or Decay} & \text{Rate} \\ \hline 30(1.25)^x & \text{growth} & 25\% \\ 500(0.75)^x & \text{decay} & 25\% \\ 2(2)^x & \text{growth} & 100\% \\ 4,000(1.01)^x & \text{growth} & 1\% \\ 7,000(0.99)^x & \text{decay} & 1\% \\ \hline \end{array} \][/tex]

This table provides the correct labels for each exponential expression, identifying whether it models growth or decay, along with the corresponding rates.
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