A hiker stands on an isolated mountain peak near sunset and observes a rainbow caused by water droplets in the air 8.00km away. The valley is 2.00km below the mountain peak and entirely flat. What fraction of the complete circular arc of the rainbow is visible to the hiker?
To determine the fraction of the complete circular arc of the rainbow that is visible to the hiker, we need to consider the geometry involved.
Let's visualize the situation. Draw a diagram with the mountain peak as point A, the valley as point B, and the position of the rainbow as point C. The distance from A to C is 8.00 km, and the distance from A to B is 2.00 km. The hiker is standing at point A.
Now, consider the right triangle formed by points A, B, and C. The base of the triangle is AC, and the height is AB. We can use the Pythagorean theorem to find the length of AC, which is the hypotenuse:
AC^2 = AB^2 + BC^2
Substituting the given values:
AC^2 = (2.00 km)^2 + (8.00 km)^2
AC^2 = 4.00 km^2 + 64.00 km^2
AC^2 = 68.00 km^2
AC ≈ 8.25 km
So, the length of AC, which represents the arc of the rainbow visible to the hiker, is approximately 8.25 km.
To find the length of the complete circular arc of the rainbow, we need to determine the radius of the circle. The radius can be found by dividing the length of AC by 2(pi):
radius = AC / (2(pi))
radius = 8.25 km / (2 * 3.14159)
radius ≈ 1.31 km
Now that we have the radius, we can calculate the length of the complete circular arc of the rainbow. The formula for the circumference of a circle is 2(pi)r:
arc length = 2(pi) * radius
arc length = 2 * 3.14159 * 1.31 km
arc length ≈ 8.23 km
Therefore, the fraction of the complete circular arc of the rainbow visible to the hiker is:
visible arc length / complete arc length
8.25 km / 8.23 km
Simplifying the fraction gives us:
approximately 1.002
So, the fraction of the complete circular arc of the rainbow visible to the hiker is approximately 1.002.