A flagpole which is 40 feet high casts a shadow on level ground. At the time when the shadow is 30 feet long, the angle that the sun makes with the horizon is changing at a rate of 15o per hour. Find the rate of change in the length of the shadow at that same time.

Question

Mathematics

Rhys Vasquez

8 June, 07:49

A flagpole which is 40 feet high casts a shadow on level ground. At the time when the shadow is 30 feet long, the angle that the sun makes with the horizon is changing at a rate of 15o per hour. Find the rate of change in the length of the shadow at that same time.

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Cedric · Answer 1

Answer: The rate = 8.5 ft/h

Step-by-step explanation:

Since we are going to need to differentiate to find the rate of change of θ we need to express it in radians rather than degrees.

Therefore, 15 degree per hour will be expressed as

15° * π/180 = 0.2618rad/hour

Using trigonometry function to find Ø

Tan Ø = 40/30 = 1.333

Ø = 53 degree

Convert it to radian

Ø = 0.93 rad

The changing at a rate of 15o per hour will be

Rate = radian / time

0.2618 = (0.93 - 0) / t

t = 0.93/0.2618

t = 3.5 hours

The rate of change in the length of the shadow at that same time will be:

Rate = 30/3.5 = 8.5 ft / hour