The shadow of a pendulum cast on a flat board moves on a straight line. By placing the x-axis on the straight line with the origin at the middle of the total path, the x-coordinate of the shadow is given by the following function: x(t) = 34cos(πt), where t is in seconds and x is in centimeters.

How much distance, in centimeters, does the shadow travel in 20 sec?

1360 cm

Explanation:

Oscillating Motion

The shadow of  a pendulum's x-coordinate is given by

x(t) = 34cos(πt)

Where 34 cm is the amplitude of the oscillation and is the angular frequency. Each time the shadow completes a whole cycle, it travels four times the amplitude: Twice from the peak to the valley and twice from the valley back to the peak. This means that each cycle the shadow travels 4*34=136 cm

Let's find the period of the oscillations. Given w, we can know the period by the formula

If we want to calculate the distance traveled by the shadow in 20 seconds, we can see it makes 20/2=10 full cycles, each one traveling 136 cm, thus the total distance traveled is 10*136=1360 cm

d. air pressure decreases as altitude increases.

answer: 31, 5556. 952 seconds