A surveyor wants to find the height of a hill. He determines that the angle of elevation to the top of the hill is 50° . He then walks 40 feet farther from the base from the hill and determines that the angle of elevation to the top of the hill is now 30° . Find the height of the hill (round to the nearest foot).
To find the height of the hill, we can use the concept of trigonometry. Let's break down the problem and understand the steps to solve it:
1. Draw a diagram: Visualize a right triangle where the hill's base forms the horizontal leg, the height of the hill forms the vertical leg, and the line of sight from the surveyor to the top of the hill forms the hypotenuse.
2. Identify the given information:
- The first angle of elevation is 50°.
- The second angle of elevation is 30°.
- The distance walked 40 feet farther from the base.
3. Determine the variables to find:
- The height of the hill (let's call it h).
4. Set up the trigonometric equations:
- In the first triangle, using the tangent function, we have:
tan(50°) = h / x
- In the second triangle, using the tangent function, we have:
tan(30°) = h / (x + 40)
Here, x represents the distance from the base of the hill to the point where the first observation was taken.
5. Find the value of x using the first equation:
- Rearrange the first equation to solve for x:
x = h / tan(50°)
6. Substitute the value of x into the second equation:
- Replace x in the second equation, we get:
tan(30°) = h / (h / tan(50°) + 40)
7. Solve for h:
- Simplify the equation:
tan(30°) = h / (h / tan(50°) + 40)
tan(30°) = tan(50°) / (1 + 40tan(50°) / h)
Note: tan(30°) and tan(50°) are known trigonometric values.
- Cross multiply and solve for h:
h * tan(30°) = tan(50°) * (h + 40tan(50°))
h * tan(30°) - tan(50°) * h = 40tan(50°) * tan(30°)
h * (tan(30°) - tan(50°)) = 40tan(50°) * tan(30°)
h = (40tan(50°) * tan(30°)) / (tan(30°) - tan(50°))
8. Finally, calculate the value of h:
- Substitute the values of tan(30°) and tan(50°), which are approximately 0.577 and 1.192, respectively, and calculate h:
h = (40 * 1.192 * 0.577) / (0.577 - 1.192)
- Evaluating the above expression, we find:
h ≈ 96 feet
Therefore, the height of the hill, rounded to the nearest foot, is approximately 96 feet.
If you draw a diagram, you can see that if the height is h, then
h cot50° - h cot30° = 40