To find the formula for the exponential function of the form $A = f(t) = I B^t$ that passes through the points $(4.7, 100)$ and $(7.7, 157)$, we need to establish two equations based on the given points.

1. From the point $(4.7, 100)$: $100 = I B^{4.7} \quad \text{(1)}$

2. From the point $(7.7, 157)$: $157 = I B^{7.7} \quad \text{(2)}$

Now, we can solve this system of equations. First, we can express $I$ from equation (1): $I = \frac{100}{B^{4.7}} \quad \text{(3)}$

Substituting equation (3) into equation (2): $157 = \left( \frac{100}{B^{4.7}} \right) B^{7.7}$

This simplifies to: $157 = 100 B^{7.7 - 4.7} = 100 B^{3}$

Now, solve for $B$: $B^3 = \frac{157}{100} = 1.57$ $B = (1.57)^{\frac{1}{3}} \approx 1.175$

Next, substitute $B$ back into equation (3) to find $I$: $I = \frac{100}{(1.175)^{4.7}}$ Calculating $(1.175)^{4.7}$: $(1.175)^{4.7} \approx 2.154$ So, $I \approx \frac{100}{2.154} \approx 46.4$

Thus, the exponential function can be expressed as: $A = f(t) \approx 46.4 \times (1.175)^t$

Summary:

The formula for the exponential function is: $A = f(t) \approx 46.4 \times (1.175)^t$

Would you like more details on any part of this process? Here are some related questions:

1. How do you find $I$ and $B$ using other methods?
2. What is the significance of the parameters $I$ and $B$?
3. How can you graph this exponential function?
4. How do exponential functions model real-world phenomena?
5. What are some properties of exponential growth?

Tip: When working with exponential functions, logarithmic transformations can be useful for linearizing the data.