To write the function $f(t) = 6700(0.96)^t$ in the form $f(t) = ab^{\frac{t}{25}}$, we need to express $0.96^t$ in the form $b^{\frac{t}{25}}$.

Step 1: Identify the equivalent base

We start by comparing the exponents in the two forms:

$0.96^t = \left(b^{\frac{1}{25}}\right)^t$

Taking the $t$-th root on both sides, we get:

$0.96 = b^{\frac{1}{25}}$

Step 2: Solve for $b$

To solve for $b$, raise both sides to the power of 25:

$b = 0.96^{25}$

Step 3: Substitute into the original form

Now, substitute the value of $b$ back into the equation:

$f(t) = 6700 \cdot \left(0.96^{25}\right)^{\frac{t}{25}}$

Thus, the function $f(t) = 6700(0.96)^t$ can be written in the form $f(t) = ab^{\frac{t}{25}}$ as:

$\boxed{f(t) = 6700 \cdot (0.96^{25})^{\frac{t}{25}}}$

Where $a = 6700$ and $b = 0.96^{25}$.

Final Answer:

$f(t) = 6700 \cdot (0.96^{25})^{\frac{t}{25}}$

Would you like more details on any part of the solution or have any questions?

Questions:

1. How do you calculate $0.96^{25}$?
2. What is the interpretation of the exponential form $f(t) = ab^{\frac{t}{25}}$?
3. How do you convert different exponential forms?
4. What is the significance of the base $0.96$ in the context of decay?
5. How does the value of $t$ affect the function $f(t)$?
6. Can this approach be generalized for any similar functions?
7. What are common applications of exponential decay in real life?
8. What if $t$ were negative—how would it affect $f(t)$?

Tip:

When dealing with exponential functions, it's useful to understand the properties of exponents, particularly when changing bases or rewriting forms for simplification.