https://file.aiquickdraw.com/mathbot/file/17303359037998k16n0dy.jpg

To solve this problem, let's set up a few variables and use related rates.

### Problem Setup and Variables
1. Let \( h = 19 \) ft be the height of the pole.
2. Let \( y = 5 \) ft be the height of the girl.
3. Let \( x \) be the distance of the girl from the pole.
4. Let \( s \) be the distance of the tip of her shadow from the pole.

The girl is moving away from the pole at a speed of \( \frac{dx}{dt} = 7 \) ft/sec. We need to find the rate at which the tip of her shadow is moving away from the pole, \( \frac{ds}{dt} \), when \( x = 48 \) ft.

### Similar Triangles
Since the pole, the girl, and the shadow form similar triangles, we can set up a ratio based on their heights and distances:

\[
\frac{h}{s} = \frac{y}{s - x}
\]

Substitute the values for \( h \) and \( y \):

\[
\frac{19}{s} = \frac{5}{s - x}
\]

Cross-multiply to eliminate the fraction:

\[
19(s - x) = 5s
\]

Expanding and rearranging terms:

\[
19s - 19x = 5s
\]

\[
14s = 19x
\]

\[
s = \frac{19}{14}x
\]

### Differentiating with Respect to Time
Now, differentiate both sides with respect to time \( t \) to find \( \frac{ds}{dt} \):

\[
\frac{ds}{dt} = \frac{19}{14} \frac{dx}{dt}
\]

### Substitute Known Values
Given \( \frac{dx}{dt} = 7 \) ft/sec, we can now calculate \( \frac{ds}{dt} \):

\[
\frac{ds}{dt} = \frac{19}{14} \times 7 = \frac{133}{14} = 9.5 \text{ ft/sec}
\]

### Answer
The rate at which the tip of her shadow is moving away from the top of the pole is \( 9.5 \) ft/sec.

Would you like further details on this solution or have any questions?

---

Here are some related questions that extend this topic:

1. What if the height of the girl or the pole changed? How would that affect the rate?
2. If the girl walks faster or slower, how would it impact the speed of her shadow?
3. What happens to the shadow length as the girl gets closer or further from the pole?
4. Can this method be used to find the shadow rate for non-constant speeds?
5. How would this problem change if the pole were on an incline?

**Tip:** In related rates problems, always set up a clear relationship between the variables before differentiating with respect to time.

A street light is at the top of a 19 ft pole. A 5 ft tall girl walks along a straight path away from the pole with a speed of 7 ft/sec. At what rate is the tip of her shadow moving away from the light when the girl is 48 ft away from the pole?

Solve a related rates problem involving a 5 ft tall girl walking away from a 19 ft pole at 7 ft/sec. Determine the rate at which the tip of her shadow moves when she is 48 ft away from the pole using similar triangles and differentiation.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation