Let's break down each problem and express them as composite functions, using $u(t)$ for the inside function and either $y(u)$ or $P(u)$ for the outside function.

(a) $y = (8t^2 - 5)^2$

Here, we want to express this equation as a composite function. The inside function is what is inside the outermost operation (in this case, the squaring operation).

• Inside function $u(t)$: This will be the expression inside the square function.
  $u(t) = 8t^2 - 5$

• Outside function $y(u)$: This represents the squaring operation applied to the inside function.
  $y(u) = u^2$

(b) $P = 8e^{-0.5t}$

Here, the goal is again to express the equation as a composite function. The inside function is what is inside the exponential function.

• Inside function $u(t)$: The exponent is the inside function.
  $u(t) = -0.5t$

• Outside function $P(u)$: The outer function is the exponential operation multiplied by 8.
  $P(u) = 8e^u$

(c) $C = 5(y^6 + 7)$

Here, we need to express the equation for $C$ as a composite function with $u(y)$ as the inside function and $C(u)$ as the outside function.

• Inside function $u(y)$: This is the expression inside the function applied to 5.
  $u(y) = y^6 + 7$

• Outside function $C(u)$: The outer function is multiplying by 5.
  $C(u) = 5u$

Summary:

1. (a)
   $u(t) = 8t^2 - 5, \quad y(u) = u^2$

2. (b)
   $u(t) = -0.5t, \quad P(u) = 8e^u$

3. (c)
   $u(y) = y^6 + 7, \quad C(u) = 5u$

Would you like further details or clarification?

---

Here are 5 related questions to expand on these concepts:

1. How would you find the derivative of $y = (8t^2 - 5)^2$ using the chain rule?
2. What is the meaning of the inside and outside functions when applying the chain rule?
3. Can you find the derivative of $P = 8e^{-0.5t}$?
4. How would the functions change if the outside function were a cube instead of a square?
5. How would you apply these concepts to a real-world problem, like population growth?

Tip: When differentiating composite functions, always remember to multiply the derivative of the outside function by the derivative of the inside function (chain rule).