Math Problem Statement
A street light is at the top of a 16ft pole. A 5ft tall girl walks along a straight path away from the pole with a speed of 5ft/sec. At what rate is the tip of her shadow moving away from the light (ie. away from the top of the pole) when the girl is 29ft away from the pole?
Solution
Let's solve this problem step-by-step using related rates.
Step 1: Label the variables
- Let be the distance of the girl from the base of the pole (in feet).
- Let be the length of the girl's shadow (in feet).
- The height of the pole is 16 feet, and the height of the girl is 5 feet.
Step 2: Set up the geometric relationship
At any given time, the triangle formed by the top of the pole, the tip of the shadow, and the base of the pole is similar to the triangle formed by the top of the girl's head, the tip of her shadow, and the base of the pole.
Using the properties of similar triangles, we have the following proportion:
Substituting the known values:
Step 3: Solve the equation for
Now, let's solve this proportion for :
Expanding:
Subtract from both sides:
Solve for :
Step 4: Differentiate with respect to time
We need to find how fast the tip of the shadow is moving, which is the rate of change of with respect to time. First, differentiate both sides of the equation with respect to time :
Given that the girl is walking at a speed of 5 ft/sec, we have:
So,
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 given by:
Substitute the known values:
To simplify:
Thus, the rate at which the tip of the shadow is moving is:
Step 6: Evaluate when
At the moment when the girl is 29 feet from the pole (), we substitute this value into the equation for to find the length of the shadow:
So, the tip of the shadow is moving at a rate of ft/sec when the girl is 29 feet away from the pole.
Final Answer:
The tip of the shadow is moving at feet per second, or approximately 7.27 ft/sec when the girl is 29 feet away from the pole.
Would you like more details on any part of the solution? Here are some related questions that might interest you:
- How would the rate of the tip of the shadow change if the girl were walking faster or slower?
- What happens if the girl walks towards the pole instead of away from it?
- How can we model this situation using other geometric principles?
- How would the rate change if the pole were taller or shorter?
- What is the exact point where the girl's shadow reaches its maximum length in relation to her position?
Tip: It's helpful to remember that when two triangles are similar, their corresponding sides are proportional—this is a key concept in solving related rates problems involving shadows!
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Math Problem Analysis
Mathematical Concepts
Related Rates
Geometry
Similar Triangles
Rate of Change
Formulas
\frac{16}{x + y} = \frac{5}{y}
y = \frac{5}{11}x
\frac{d}{dt}(x + y) = \frac{dx}{dt} + \frac{dy}{dt}
Theorems
Similar Triangles
Suitable Grade Level
Grades 11-12
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