A civil engineer is designing a road that will pass through a tunnel. The tunnel opening is a semi-circle with a diameter of 20 feet. If the road needs to be 16 feet wide at the base of the tunnel, what is the maximum height of a vehicle that can pass through the tunnel at the edge of the 16-foot wide road?
- A6 feet
- B8 feet
- C10 feet
- D12 feet
Show answer & explanationAnswer & explanation
Correct answer: A. 6 feet
The diameter of the semi-circular tunnel is 20 feet, so its radius (r) is 10 feet. Imagine a coordinate plane with the center of the semi-circle at the origin (0,0). The equation of the semi-circle is x² + y² = r². The road is 16 feet wide, so its edges are at x = -8 and x = 8 (since the total width is 16, half is 8 from the center). We need to find the height (y) when x = 8. Using the circle equation: 8² + y² = 10². This gives 64 + y² = 100. So, y² = 100 - 64 = 36. Therefore, y = √36 = 6 feet. This represents the maximum height at the edge of the 16-foot wide road.
Why the other options are wrong
- B. This is the half-width of the road (16/2), not the height.
- C. This is the radius of the tunnel, which is the height at the very center of the road.
- D. Incorrect calculation; possibly 10 + 2 for some reason.
Circle Equation Application
Using the standard equation of a circle (x-h)² + (y-k)² = r² to solve for unknown dimensions within a circular context.
- For a circle centered at the origin (0,0), the equation simplifies to x² + y² = r².
- This is directly related to the Pythagorean theorem.
- Can be used to find a coordinate (height or width) given another coordinate and the radius.
Memory trick: X squared, plus Y squared, equals R squared, for a circle at the center, it's declared!