Semi- Circles (Tangents?)

Question

Inside a semi- circle tunnel with a diameter of 30 feet, a vertical support beam is placed 8 feet from the side of the tunnel. How tall is the beam?

Answer 1

First, note that the beam is placed 30/2 - 8 = 7 feet from the center of the circular tunnel. We can use Pythagorean Theorem to calculate the length of the beam.

Picture the center of the circle, the bottom point of the beam, and one of the points where the beam intercepts the circle as forming a right triangle. Let the height of the beam be m. Then,

m^2+7^2 = (30/2)^2 = 15^2.

m = sqrt(15^2 - 7^2) = sqrt(176).

Therefore, the beam is sqrt(176) feet tall.

Answer 2

x² + y² = 15²

7² + y² = 225

y² = 225 - 49

y = ±√176

Ignore the negative.

y = 4√11