Let's solve the three parts of the problem step by step.

a) Calculate $M$ with $w = 2$ and $p = 3.5$

The formula given is:

$M = wp + 6$

Substituting $w = 2$ and $p = 3.5$:

$M = 2 \times 3.5 + 6$

Calculate:

$M = 7 + 6 = 13$

So, $M = 13$.

b) Rearrange the formula to solve for $p$

We start with the given formula:

$M = wp + 6$

To solve for $p$, we first subtract 6 from both sides:

$M - 6 = wp$

Now, divide both sides by $w$:

$p = \frac{M - 6}{w}$

So the formula for $p$ is:

$p = \frac{M - 6}{w}$

c) Calculate $p$ for $M = 15$ and $w = 3$

Using the formula we derived in part b:

$p = \frac{M - 6}{w}$

Substituting $M = 15$ and $w = 3$:

$p = \frac{15 - 6}{3}$

Calculate:

$p = \frac{9}{3} = 3$

So, $p = 3$.

Would you like more detailed explanations for any part of these calculations? Here are some related questions for further exploration:

1. What if $w$ or $p$ were negative? How would that affect the value of $M$?
2. How can we graph the function $M = wp + 6$ for different values of $w$ and $p$?
3. What is the significance of the constant $6$ in the formula?
4. How would you solve $M$ if given an inequality involving $w$ and $p$?
5. Could we interpret $M, w,$ and $p$ in a practical, real-world context?

Tip: When dealing with algebraic formulas, it's helpful to isolate the desired variable step-by-step by performing inverse operations in the correct order.