To solve the problem of determining how fast the tip of the man's shadow is moving, we will use related rates in calculus. Let's set up the problem with the necessary variables and relationships.

First, define the variables:

• Let $y$ be the length of the man's shadow in feet.
• Let $x$ be the distance between the man and the pole in feet.
• Let $L$ be the distance from the tip of the shadow to the pole, $L = x + y$.

Given:

• The height of the pole is 15 feet.
• The height of the man is 6 feet.
• The man walks away from the pole at a speed of 4 ft/s, so $\frac{dx}{dt} = 4$ ft/s.

Using similar triangles, we can set up the following proportion based on the heights and lengths of the shadows: $\frac{15}{L} = \frac{6}{y}$

Solving for $L$ in terms of $y$: $15y = 6L \implies L = \frac{15}{6}y \implies L = \frac{5}{2}y$

Since $L = x + y$: $x + y = \frac{5}{2}y \implies x = \frac{5}{2}y - y \implies x = \frac{3}{2}y$

Differentiate both sides with respect to time $t$: $\frac{d}{dt}(x) = \frac{d}{dt}\left(\frac{3}{2}y\right)$ $\frac{dx}{dt} = \frac{3}{2} \frac{dy}{dt}$

We know $\frac{dx}{dt} = 4$ ft/s, so: $4 = \frac{3}{2} \frac{dy}{dt}$

Solving for $\frac{dy}{dt}$: $\frac{dy}{dt} = \frac{4 \cdot 2}{3} = \frac{8}{3} \text{ ft/s}$

Thus, the speed at which the tip of the shadow is moving is: $\frac{dL}{dt} = \frac{dx}{dt} + \frac{dy}{dt} = 4 + \frac{8}{3} = \frac{12}{3} + \frac{8}{3} = \frac{20}{3} \text{ ft/s}$

So, the tip of the shadow is moving at a rate of $\frac{20}{3}$ ft/s when the man is 40 feet from the pole.

Would you like further details or have any questions?

Here are 8 related questions you might consider:

1. What is the length of the man's shadow when he is 40 feet away from the pole?
2. How would the speed of the shadow's tip change if the man walked at a different speed?
3. How is the rate of change of the shadow's length affected by the height of the pole?
4. What happens to the rate of change of the shadow's length if the man's height is different?
5. How would you set up a similar problem for a different height of the pole?
6. How does the concept of similar triangles help in solving this problem?
7. What if the man were walking towards the pole instead? How would that affect the rates?
8. Can you explain why we use the differentiation approach to solve this problem?

Tip: Always double-check your related rates setup by ensuring the units of measurement are consistent throughout the problem.