Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![**Exponential Function Determination**
**Task:**
Use the given information to find a formula for the exponential function \( N = N(t) \).
**Given Values:**
- \( N(3) = 9 \)
- \( N(5) = 1 \)
**Objective:**
Find the function \( N(t) \).
**Steps to Solve:**
1. **Set up the equation**: Use the general form of an exponential function, \( N(t) = ab^t \).
2. **Use given points**:
- For \( N(3) = 9 \):
\[
ab^3 = 9
\]
- For \( N(5) = 1 \):
\[
ab^5 = 1
\]
3. **Calculations**:
- Solving the ratio \( \frac{N(3)}{N(5)} = \frac{9}{1} \) gives a ratio which represents a way to find the base of the exponential function:
\[
\frac{b^3}{b^5} = \frac{9}{1} \Rightarrow b^{-2} = \frac{1}{9} \Rightarrow b = \left(\frac{1}{9}\right)^{-\frac{1}{2}} = 3
\]
- Thus, substituting back, we use \( ab^3 = 9 \) with \( b = 3 \):
\[
a \cdot 27 = 9 \Rightarrow a = \frac{1}{3}
\]
4. **Function Formulation**:
Therefore, the exponential function is:
\[
N(t) = \frac{1}{3} \cdot 3^t
\]
This solution demonstrates the step-by-step process of finding the exponential function that fits given data points by identifying \( a \) and \( b \) in the formula \( N(t) = ab^t \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc49c9615-4f22-4f84-a617-639a09f12ba4%2F1fcec086-b9fb-429c-9ad8-559ad46bb5e2%2Fw0r70np_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exponential Function Determination**
**Task:**
Use the given information to find a formula for the exponential function \( N = N(t) \).
**Given Values:**
- \( N(3) = 9 \)
- \( N(5) = 1 \)
**Objective:**
Find the function \( N(t) \).
**Steps to Solve:**
1. **Set up the equation**: Use the general form of an exponential function, \( N(t) = ab^t \).
2. **Use given points**:
- For \( N(3) = 9 \):
\[
ab^3 = 9
\]
- For \( N(5) = 1 \):
\[
ab^5 = 1
\]
3. **Calculations**:
- Solving the ratio \( \frac{N(3)}{N(5)} = \frac{9}{1} \) gives a ratio which represents a way to find the base of the exponential function:
\[
\frac{b^3}{b^5} = \frac{9}{1} \Rightarrow b^{-2} = \frac{1}{9} \Rightarrow b = \left(\frac{1}{9}\right)^{-\frac{1}{2}} = 3
\]
- Thus, substituting back, we use \( ab^3 = 9 \) with \( b = 3 \):
\[
a \cdot 27 = 9 \Rightarrow a = \frac{1}{3}
\]
4. **Function Formulation**:
Therefore, the exponential function is:
\[
N(t) = \frac{1}{3} \cdot 3^t
\]
This solution demonstrates the step-by-step process of finding the exponential function that fits given data points by identifying \( a \) and \( b \) in the formula \( N(t) = ab^t \).
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