11-1 Lines that Intersect Circles
15. Olympus Mons's peak rises 25 kilometers above the surface of the planet Mars. The diameter of Mars is approximately 6,794 kilometers. What is the distance from the peak of Olympus Mons to the horizon to the nearest kilometer?
The answer is 413 km.
413
To find the distance from the peak of Olympus Mons to the horizon, we can use the Pythagorean theorem. Let's assume that the peak of Olympus Mons is at the center of a circle, and the radius of the circle is the distance from the peak to the horizon. We can draw a line from the peak to the edge of the circle, which will be the radius.
First, we need to find the radius of the planet Mars. The radius is half of the diameter, so the radius of Mars is 6,794 kilometers / 2 = 3,397 kilometers.
Now, we have created a right triangle with the radius (3,397 kilometers) as one leg and the height of Olympus Mons (25 kilometers) as the other leg. The hypotenuse of the triangle represents the distance from the peak to the horizon.
Using the Pythagorean theorem, we can find the hypotenuse (H) of the triangle:
H^2 = (leg1)^2 + (leg2)^2
H^2 = (3,397 kilometers)^2 + (25 kilometers)^2
H^2 = 11,519,609 kilometers^2 + 625 kilometers^2
H^2 = 11,520,234 kilometers^2
To find the value of H, we take the square root of both sides:
H = √(11,520,234 kilometers^2)
H ≈ 3,395.5 kilometers
Therefore, the distance from the peak of Olympus Mons to the horizon is approximately 3,395.5 kilometers, which can be rounded to 3,396 kilometers to the nearest kilometer.