To solve this related rates problem, we need to analyze the situation step-by-step.

---

Step 1: Variables and Relationships

1. Define the variables:

   • $x$: The distance between the man and the wall (in feet).
   • $y$: The height of the shadow on the wall (in feet).
   • $h$: The height of the man ($h = 6$ ft, constant).
   • $dx/dt = -2 \, \text{ft/sec}$: The rate at which the man is moving toward the wall (negative because $x$ is decreasing).

2. Geometry of the situation:

   • The spotlight, the top of the man, and the top of the shadow form similar triangles.
   • The first triangle is formed by the spotlight, the wall, and the shadow height ($y$).
   • The second triangle is formed by the spotlight, the ground, and the man's height ($h$).

---

Step 2: Set up the proportional relationship

Using the property of similar triangles: $\frac{y}{21} = \frac{h}{x}$ Substitute $h = 6$: $\frac{y}{21} = \frac{6}{x}$

Simplify: $y = \frac{126}{x}$

---

Step 3: Differentiate with respect to time ($t$)

Differentiate $y = \frac{126}{x}$ implicitly with respect to $t$: $\frac{dy}{dt} = -\frac{126}{x^2} \cdot \frac{dx}{dt}$

Substitute $\frac{dx}{dt} = -2 \, \text{ft/sec}$: $\frac{dy}{dt} = -\frac{126}{x^2} \cdot (-2)$ $\frac{dy}{dt} = \frac{252}{x^2}$

---

Step 4: Solve for $\frac{dy}{dt}$ when $x = 6$ ft

Substitute $x = 6$: $\frac{dy}{dt} = \frac{252}{6^2}$ $\frac{dy}{dt} = \frac{252}{36} = 7 \, \text{ft/sec}$

---

Final Answer

The height of the shadow on the wall is increasing at a rate of: $\boxed{7 \, \text{ft/sec}}$

---

Would you like a step-by-step explanation of any part of the solution?
Here are 5 related questions for deeper understanding:

1. How does the principle of similar triangles apply in this scenario?
2. Why is the rate of change of $x$ negative?
3. What happens to the rate of change of $y$ as the man gets closer to the wall?
4. Can this problem be solved using other methods besides similar triangles?
5. How does the spotlight's distance from the wall affect the problem?

Tip: Always label variables clearly when solving related rates problems to avoid confusion!