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Canyon Hiker Distance Calculation with Angles of Depression (37° and 21°)
Mathematics
Grade 10 (Junior High School)
Question Content
Two hikers on opposite sides of a canyon each stand precisely 525 meters above the canyon floor. They each sight a landmark on the canyon floor on a line directly between them. The angles of depression from each hiker to the landmark meter are 37° and 21°. How far apart are the hikers? Round your answer to the nearest whole meter.
Correct Answer
2016 meters
Detailed Solution Steps
1
Step 1: Recognize that the angle of depression is equal to the angle of elevation from the landmark to the hiker (alternate interior angles theorem). So we have two right triangles, each with a vertical side of 525 meters, and angles of elevation 37° and 21° respectively.
2
Step 2: For the triangle with the 37° angle, use the tangent function: $\\tan(\\theta) = \\frac{opposite}{adjacent}$. Let $x_1$ be the horizontal distance from this hiker to the landmark. So $\\tan(37°) = \\frac{525}{x_1}$, which rearranges to $x_1 = \\frac{525}{\\tan(37°)}$. Calculating this, $\\tan(37°) \\approx 0.7536$, so $x_1 \\approx \\frac{525}{0.7536} \\approx 697$ meters.
3
Step 3: For the triangle with the 21° angle, let $x_2$ be the horizontal distance from this hiker to the landmark. Using the tangent function again: $\\tan(21°) = \\frac{525}{x_2}$, which rearranges to $x_2 = \\frac{525}{\\tan(21°)}$. Calculating this, $\\tan(21°) \\approx 0.3839$, so $x_2 \\approx \\frac{525}{0.3839} \\approx 1367$ meters.
4
Step 4: The total distance between the two hikers is $x_1 + x_2$. Adding the two values: $697 + 1367 = 2064$? Correction: $x_1 = 525 / tan37 ≈ 525 / 0.75355 ≈ 696.7$, $x_2 = 525 / tan21 ≈ 525 / 0.38386 ≈ 1367.7$. Sum is $696.7 + 1367.7 ≈ 2064.4$, rounded to nearest whole number is 2064 meters. (The original draft calculation had an error in variable definition; correct approach uses adjacent side as horizontal distance, so tangent is opposite over adjacent where opposite is height)
5
Corrected Step 2 & 3: Actually, angle of depression forms a right triangle where height is opposite side, horizontal distance is adjacent. So $tan(angle) = \\frac{height}{horizontal distance}$ → horizontal distance = $\\frac{height}{tan(angle)}$. So $x1 = 525 / tan(37°) ≈ 697$, $x2 = 525 / tan(21°) ≈ 1367$. Sum is 697 + 1367 = 2064 meters.
Knowledge Points Involved
1
Angle of Depression and Elevation Relationship
The angle of depression from a higher point to a lower point is equal to the angle of elevation from the lower point to the higher point, due to alternate interior angles formed by a horizontal transversal line cutting two parallel horizontal lines. This is used to convert the given angle of depression to an angle we can use in the right triangle on the canyon floor.
2
Tangent Function in Right Triangles
For an acute angle in a right triangle, $\\tan(\\theta) = \\frac{opposite side}{adjacent side}$. It relates the angle of a right triangle to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. This is used to calculate unknown horizontal distances when we know the vertical height and the angle of elevation.
3
Right Triangle Problem-Solving for Real-World Scenarios
Real-world distance problems involving heights and angles can be modeled using right triangles. We break the scenario into separate right triangles, apply trigonometric functions to find unknown side lengths, then combine the results to find the total desired distance.
4
Rounding Decimal Values to Whole Numbers
When a problem asks for a rounded whole number result, we calculate the precise decimal value first, then round it to the nearest integer by looking at the tenths place: if it is 5 or greater, we round up, otherwise we round down.
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