Question

Related Rates

A cylindrical tank with radius 5 m is being filled with water
at a rate of 3 m/min. How fast is the height of the water
increasing?
A street light is mounted at the top of a 15-ft-tall pole. A man
6 ft tall walks away from the pole with a speed of 5 ft/s along
a straight path. How fast is the tip of his shadow moving when
he is 40 ft from the pole?​

Answer 1

1. 0.382 m/min
2. 8 1/3 ft/s

Explanation:

1. The rate of change of volume is ...

V = πr^2h

V' = πr^2h'

h' = V'/(πr^2) = (3 m^3/min)/(π(5 m)^2) = 0.12/π m/min

h' ≈ 0.382 m/min

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2. In the attached, ∆PCM ~ ∆MXS, so we can write the proportion ...

x/9 = s/15 . . . . where x = BX and s = BS in the diagram

Multiplying by 15 gives ...

s = (15/9)x = (5/3)x

Then the rate of change is ...

s' = (5/3)x' = (5/3)(5 ft/s)

s' = 8 1/3 ft/s

The tip of the shadow is moving at 8 1/3 feet per second.