https://file.aiquickdraw.com/mathbot/file/1730248164812q48je6ao.jpg

The problem requires identifying the initial value and the growth factor of an exponential function given in the form \( Q(t) = ab^t \).

The function provided is:
\[
Q(t) = \frac{25}{3^{\frac{1}{3}}}
\]

To rewrite this in the form \( Q = ab^t \), we’ll analyze it as follows:

1. **Identify the Initial Value \( a \):**
   The initial value is the value of \( Q \) when \( t = 0 \).
   Substituting \( t = 0 \):
   \[
   Q(0) = \frac{25}{3^{0}} = 25
   \]
   So, the initial value \( a \) is **25**.

2. **Determine the Growth Factor \( b \):**
   In an exponential function of the form \( Q(t) = ab^t \), \( b \) represents the growth factor. From the equation, we see that \( b = 3^{\frac{1}{3}} \).

   Calculating \( 3^{\frac{1}{3}} \) to four decimal places:
   \[
   3^{\frac{1}{3}} \approx 1.4422
   \]

So, the answers are:
- **The initial value** is \( 25 \).
- **The growth factor** is approximately \( 1.4422 \).

Would you like a detailed explanation of each step?

---

**Follow-up Questions:**

1. What is an exponential function, and why does it have a constant growth factor?
2. How do you interpret a growth factor that is greater than 1 versus one that is less than 1?
3. How can you convert an exponential decay function to a growth function?
4. What is the significance of the initial value in an exponential function in real-world applications?
5. How do we calculate fractional exponents, like \( 3^{1/3} \), without a calculator?

**Tip:** To better understand exponential functions, try graphing different functions and observing the effect of changes in the initial value and growth factor.

Write an exponential function in the form Q = ab^t and identify the initial value and the growth factor.

Given:
Q(t) = 25 / 3^(1/3)

State the initial value and the growth factor. Note: Round your answer for the growth factor to four decimal places.

Find the initial value and growth factor for the exponential function Q(t) = 25 / 3^(1/3). This problem involves identifying key components of exponential functions in the form Q = ab^t, focusing on algebraic manipulation and exponential growth. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation