To solve this problem, let's break it down step by step.

Problem Breakdown:

1. The function $f(a)$ increases by a factor of 27 each time $a$ increases by a constant $c$.
2. This implies that the function is likely an exponential function of the form: $f(a) = r \cdot b^{\frac{a}{c}}$ where $r$ is a coefficient and $b$ is the base.

Analysis:

• Given that each time $a$ increases by $c$, the function $f(a)$ is multiplied by 27: $f(a + c) = 27 \cdot f(a)$

  Substituting the exponential form: $r \cdot b^{\frac{a+c}{c}} = 27 \cdot r \cdot b^{\frac{a}{c}}$

  Simplify the exponent on the left: $r \cdot b^{\frac{a}{c} + 1} = 27 \cdot r \cdot b^{\frac{a}{c}}$

  Since $r$ and $b^{\frac{a}{c}}$ are common factors on both sides, we can cancel them out: $b = 27$

  Therefore, the function can be written as: $f(a) = r \cdot 27^{\frac{a}{c}}$

Conclusion:

The equivalent form of the function $f(a)$ that shows $r$ as a coefficient is: $f(a) = r \cdot 27^{\frac{a}{c}}$

This form clearly displays $r$ as the coefficient of $27^{\frac{a}{c}}$.

Follow-Up

Do you want more details or have any questions about the solution?

Here are some related questions for further exploration:

1. What is the impact on the function if the factor of increase was different from 27?
2. How would the function change if $f(a)$ decreased by a factor of 27 instead of increasing?
3. What is the derivative of this function with respect to $a$?
4. How does the base 27 relate to the logarithmic form of the function?
5. What happens to $f(a)$ as $a$ approaches infinity?

Tip: Exponential functions often model scenarios where quantities grow or shrink by a consistent percentage or factor over time, such as compound interest or population growth.