a hiker stands on an isolated mountain peak near sunset and observes a rainbow caused by water droplets in the air 8 km away. the valley is 2 km below the mountain peak and entirely flat. what fraction of the complete circular arc of the rainbow is visible to the hike? solution.pdf
To find the fraction of the complete circular arc of the rainbow visible to the hiker, we can use some geometry.
Let's denote the radius of the complete circular arc as "R" and the distance from the hiker to the center of the arc as "d."
Given that the hiker is standing 2 km above the valley and observing the rainbow 8 km away, we can form a right-angled triangle with the hypotenuse as the line connecting the hiker to the center of the arc.
Using the Pythagorean theorem, we can determine the value of "d":
d^2 = (8 km)^2 + (2 km)^2
d^2 = 64 km^2 + 4 km^2
d^2 = 68 km^2
d = √68 km
d ≈ 8.2462 km
Now, we can find the angle that the visible portion of the rainbow subtends at the center by dividing the distance from the hiker to the center (d) by the radius of the arc (R):
sinθ = (d / R)
To find the fraction of the complete circular arc visible to the hiker, we need to calculate the angle θ in radians and divide it by 2π (i.e., the total angle of a circle in radians).
θ / (2π) = (d / R) [Equation 1]
To find the value of θ, we can use the inverse sine function:
θ = arcsin(d / R)
Substituting this value of θ into Equation 1, we get:
arcsin(d / R) / (2π) = (d / R)
Multiplying both sides by (2π) and rearranging, we can get:
arcsin(d / R) = (d / R) * 2π
d / R = sin[(d / R) * 2π]
d / R ≈ 8.2462 / R
Now, we can solve for the desired fraction of the complete circular arc visible to the hiker:
Fraction of visible arc = (θ / 2π) = (d / R) ≈ 8.2462 / R
Please note that without the actual value of R provided, we cannot provide an exact fraction. However, the expression 8.2462 / R gives the ratio in terms of the unknown radius R.
To find the fraction of the complete circular arc of the rainbow that is visible to the hiker, we need to understand the geometry of the situation.
Let's consider the diagram:
```
+-------------------+
| |
| |
| |
| ↑ |
| | |
| |h |
| | |
+-------- M---------+----→
s r
```
M is the mountain peak where the hiker is standing, and it is 2 km above the valley. The hiker observes a rainbow caused by water droplets in the air, located 8 km away in the valley. The distance from M to the rainbow is represented by 'r', and the length of the visible arc of the rainbow is represented by 's'. We need to find the fraction s/r.
To solve the problem, we can use trigonometry and the concept of similar triangles.
Here's the step-by-step solution:
Step 1: Draw a triangle connecting the hiker (M), the center of the rainbow's circle (O), and one end of the visible arc of the rainbow (A).
```
+-------------------+
| |
| |
| (O)--------→
| | /
| | /
| ↑ | /
| | | /
+-------- M---------+
```
Step 2: Since the rainbow is caused by water droplets in the air, the center of the rainbow's circle (O) will be located on a line perpendicular to the valley plane, passing through the hiker (M).
```
+-------------------+
| |
| |
| (O)--------→
| | | /
| | | /
| ↑ | O/
| | | /|
+-------- M---------+
```
Step 3: The triangle MOA is similar to triangle MBA (or triangle MBA is an isosceles triangle). This is because ∠MOA is a central angle, and ∠MAB (or ∠MBA) is an inscribed angle that intercepts the same arc, which is s/r of the complete circular arc.
```
+-------------------+
| |
| (B)--------(O)--------→
| | \ | /
| | \ | /
| | ∠MOA \ |/
| | |
+-------- M----------+
```
Step 4: Using the similar triangles, we can write the following proportion:
```
MO / MA = MB / MA
```
Step 5: Since MB = r (distance from M to O), and MA = s (visible arc),
```
MO / s = r / s
```
or,
```
MO = r
```
Step 6: Applying the Pythagorean theorem to triangle MOA,
```
MA^2 = MO^2 + OA^2
```
Substituting the values,
```
s^2 = r^2 + OA^2
```
Step 7: Solving for OA,
```
OA^2 = s^2 - r^2
```
Step 8: We can calculate OA using the given information:
```
OA^2 = s^2 - r^2
OA^2 = (8^2) - (2^2)
OA^2 = 64 - 4
OA^2 = 60
OA = √60
```
Step 9: Now that we have OA, we can calculate the fraction of the complete circular arc that is visible to the hiker:
```
Fraction = (visible arc length) / (complete arc length)
Fraction = s / (2πr)
Fraction = s / (2π * r)
Fraction = s / (2π * √60)
```
This is the fraction of the complete circular arc of the rainbow that is visible to the hiker.