Question

answer the following true or false:

if an ice skater is skating directly away from a 6 meter tall lamp at 11 meters per second, and you want to calculate the rate at which the skaters shadow is increasing in length, then you need to know how tall the skater is but you do not need to know how far the skater has skated.

true
false

Explanation:

Set up the geometric model

Using the Related Rates knowledge point
Let \(H = 6\) be the lamp height, \(h\) be the skater's height, \(x\) be the distance from the lamp to the skater, and \(s\) be the length of the shadow.
By similar triangles:

$$ \frac{s}{h} = \frac{x + s}{H} $$

Differentiate with respect to time

Using the Related Rates knowledge point

$$
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$$

Analyze the required variables

Using the Related Rates knowledge point

$$ \frac{ds}{dt} = \frac{h}{H - h} \frac{dx}{dt} $$

To find \(\frac{ds}{dt}\), we need the skater's height \(h\) and speed \(\frac{dx}{dt} = 11\). We do not need the distance \(x\). Thus, the statement is True.

Answer:

• True (Correct answer)
• False