I) A Light is on Top of a 12 foot pole and an 5 fT 6 in tall person is walking away from the pole at a rate of $$ 2 \mathrm{fT} / \mathrm{sec} $$ a) What rate is the Tip of the shadow moving away from the pole when the person is 25 fT from the pole? b) AT what rate is the Tip of the shadow moving away from the person when the person is 25 fT from the pole?

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Let $$ x $$ be the distance from the pole to the person and $$ y $$ be the distance from the pole to the tip of the shadow. The person's shadow length is $$ s = y - x $$.

**Step 1. Set up the Similar Triangles**

The tall pole has height $$ 12 $$ ft and the person has height $$ 5.5 $$ ft (since $$ 5 $$ ft $$ 6 $$ in $$ = 5 + 0.5 = 5.5 $$ ft). By similar triangles, we have:
$$
\frac{12}{y} = \frac{5.5}{y-x}
$$
Cross multiply to get:
$$
12(y-x) = 5.5y
$$

**Step 2. Solve for $$ y $$ in Terms of $$ x $$**

Expand and rearrange:
$$
12y - 12x = 5.5y \quad \Longrightarrow \quad 12y - 5.5y = 12x 
$$
$$
(12 - 5.5)y = 12x \quad \Longrightarrow \quad 6.5\,y = 12x
$$
Thus,
$$
y = \frac{12}{6.5}x = \frac{24}{13}x
$$

**Step 3. Differentiate with Respect to Time**

Differentiate both sides with respect to time $$ t $$:
$$
\frac{dy}{dt} = \frac{24}{13}\frac{dx}{dt}
$$
Given that $$\frac{dx}{dt} = 2 \, \text{ft/sec}$$, we substitute:
$$
\frac{dy}{dt} = \frac{24}{13} \times 2 = \frac{48}{13} \, \text{ft/sec}
$$

**(a) Rate at Which the Tip of the Shadow is Moving Away from the Pole**

The rate is given by:
$$
\frac{dy}{dt} = \frac{48}{13} \, \text{ft/sec}
$$

**(b) Rate at Which the Tip of the Shadow is Moving Away from the Person**

The distance from the person to the tip of the shadow is $$ y - x $$. Its rate of change is:
$$
\frac{d}{dt}(y-x) = \frac{dy}{dt} - \frac{dx}{dt}
$$
Substitute the known rates:
$$
\frac{d}{dt}(y-x) = \frac{48}{13} - 2
$$
Express $$ 2 $$ as $$ \frac{26}{13} $$ to combine:
$$
\frac{d}{dt}(y-x) = \frac{48 - 26}{13} = \frac{22}{13} \, \text{ft/sec}
$$

**Final Answers:**

- (a) The tip of the shadow is moving away from the pole at a rate of $$ \frac{48}{13} $$ ft/sec.
- (b) The tip of the shadow is moving away from the person at a rate of $$ \frac{22}{13} $$ ft/sec.
When the person is 25 feet from the pole: - The tip of the shadow is moving away from the pole at $$ \frac{48}{13} $$ ft/sec. - The tip of the shadow is moving away from the person at $$ \frac{22}{13} $$ ft/sec.

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