Question

Galileo was able to estimate the height of mountains on the Moon by measuring the length of their shadows.

a. If a shadow is 5 kilometers long when the Sun is 10° above the horizon, estimate how tall the mountain is._______ km

b. Estimate at what angle the Sun must be above the horizon for the shadow to be five times as long as the height of the mountain. ____°

[Hint: To help visualize the problem, draw a tall mountain sitting on a flat plane viewed from the side. With a protractor, draw a line with a 10° angle to the plane that touches the mountain peak. This shows how long the shadow is when the Sun is 10° above the horizon.]

Answer 1

a)
tan = H/S
Where is the angle above the horizon, H is the height of the mountain, and S is the length of the shadow.
Substituting values,
tan(10) = H / 5
H = 5 * tan(10)
= 5 * 0.176
= 0.88 km

b)
tan = H/S
Where is the angle above the horizon, H is the height of the mountain, and S is the length of the shadow.
Given that S = 5 * H
Substituting,
tan = H / (5 * H)
= 1/5
= 0.2
= tan^-1(0.2)
= 11.3 degrees