Question
14) An exponential function can be modeled by the function \( P=2(1.45)^{x} \). Which of the following statements is true? Select all that apply. A. \( \square \) This function models exponential growth. B. \( \square \) This function models exponential decay. C. \( \square \) The initial amount is 0.45 . D. \( \square \) The initial amount is 2 . E. \( \square \) The rate is \( 1.45 \% \). F. \( \square \) The rate is \( 45 \% \).
Ask by Tran Hilton. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The function \( P = 2(1.45)^x \) models exponential growth, the initial amount is 2, and the rate is 45%.
Solution
Let's analyze the exponential function \( P = 2(1.45)^x \) and evaluate each statement:
**A. This function models exponential growth.**
- **True.**
The base of the exponential function is \( 1.45 \), which is greater than 1. This indicates that the function represents exponential growth.
**B. This function models exponential decay.**
- **False.**
Exponential decay would require the base to be between 0 and 1. Since \( 1.45 > 1 \), it does not model decay.
**C. The initial amount is 0.45.**
- **False.**
The initial amount is determined by the value of \( P \) when \( x = 0 \).
\( P(0) = 2(1.45)^0 = 2 \times 1 = 2 \).
**D. The initial amount is 2.**
- **True.**
As calculated above, the initial amount \( P(0) \) is 2.
**E. The rate is \( 1.45\% \).**
- **False.**
The rate is represented by \( r \) in the expression \( P = P_0(1 + r)^x \). Here, \( 1 + r = 1.45 \), so \( r = 0.45 \) or \( 45\% \).
**F. The rate is \( 45\% \).**
- **True.**
As calculated above, the rate \( r = 45\% \).
**Summary of True Statements:**
- **A. This function models exponential growth.**
- **D. The initial amount is 2.**
- **F. The rate is \( 45\% \).**
Answer: All statements A, D, and F are true.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
The function \( P=2(1.45)^{x} \) indeed models exponential growth because the base of the exponential term (1.45) is greater than 1. This indicates that as \( x \) increases, \( P \) will become larger and larger, demonstrating an increase over time. Furthermore, the initial amount, also called the y-intercept when \( x=0 \), can be found by plugging zero into the equation, giving us \( P = 2 \cdot (1.45)^0 = 2 \). To clarify the growth rate, since we multiply by 1.45, it means there is a 45% increase for each unit increase in \( x \). So, when considering the rate of growth, the rate is indeed 45%, not 1.45%. So we can confidently tick options A, D, and F.
