This is a related rates problem. Let's break it down step by step.

Step 1: Define variables

• Let $x$ be the distance (in feet) from the man to the pole.
• Let $s$ be the length of the shadow of the man.
• The height of the pole is 15 ft, and the height of the man is 6 ft.
• The man is moving away from the pole at a rate of 5 ft/sec, so $\frac{dx}{dt} = 5$ ft/sec.
• We need to find $\frac{d}{dt} (x + s)$, the rate at which the tip of the shadow is moving when $x = 40$ feet.

Step 2: Set up a proportion using similar triangles

At any given time, the top of the pole, the tip of the shadow, and the man form two similar triangles. One triangle is formed by the pole and the tip of the shadow, and the other is formed by the man and the tip of his shadow. The ratios of corresponding sides in these triangles must be equal.

$\frac{\text{height of pole}}{\text{distance from pole to tip of shadow}} = \frac{\text{height of man}}{\text{length of shadow}}$

Substitute the known values:

$\frac{15}{x + s} = \frac{6}{s}$

Step 3: Solve for $s$

Cross-multiply to solve for $s$:

$15s = 6(x + s)$

Distribute:

$15s = 6x + 6s$

Now, subtract $6s$ from both sides:

$9s = 6x$

Solve for $s$:

$s = \frac{2}{3}x$

Step 4: Differentiate with respect to time

Now that we have $s = \frac{2}{3}x$, we can differentiate both sides of this equation with respect to time $t$:

$\frac{ds}{dt} = \frac{2}{3} \frac{dx}{dt}$

We are given $\frac{dx}{dt} = 5$ ft/sec, so:

$\frac{ds}{dt} = \frac{2}{3} \times 5 = \frac{10}{3} \text{ ft/sec}$

Step 5: Find the rate at which the tip of the shadow is moving

The rate at which the tip of the shadow is moving is the sum of the rates at which the man is moving and the rate at which the shadow is lengthening:

$\frac{d}{dt}(x + s) = \frac{dx}{dt} + \frac{ds}{dt}$

Substitute the values:

$\frac{d}{dt}(x + s) = 5 + \frac{10}{3} = \frac{15}{3} + \frac{10}{3} = \frac{25}{3} \text{ ft/sec}$

Final Answer:

The tip of the shadow is moving at a rate of $\frac{25}{3}$ ft/sec, or approximately 8.33 ft/sec, when the man is 40 feet from the pole.

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Would you like more details on any of the steps, or do you have other questions?

Related Questions:

1. How would the problem change if the height of the pole was different?
2. How can you apply the concept of similar triangles to solve other real-world problems?
3. What happens if the man walks towards the pole instead of away from it?
4. How would you approach related rates problems involving circular motion?
5. How can you verify the correctness of your related rates solution?

Tip:

In related rates problems, always start by defining variables for all quantities changing with time. Use relationships between these variables (like similar triangles) to connect them, and then differentiate with respect to time.