A kite 100 ft above the ground moves horizontally at a speed of10 ft/s. At what rate is the angle (inradians) between the string and the horizontal decreasing when 200ft of string have been let out? (Enter your answer as afraction.)

Step-by-step explanation:

We can model the situation with the diagram below.

The relationship between the angle between the string and the horizontal can be modeled by:

Differentiate the equation with respect to time t:

When 200 ft of string have been left out, a = 200.

By the Pythagorean Theorem:

Then it follows that:

And since the kite moves horizontally at a speed of 10 ft/s, dx/dt = 10.

Substitute:

So:

f(x) = - 2/3x + 5; f(5/2)

substitute 5/2 in for x:

f(5/2) = -2/3(5/2) + 5

simplify the right side:

f(5/2) = -5/3 + 5

f(5/2) = 10/3

answer:

f(5/2) = 10/3

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