From the top of a house 80m high, the angles of elevation
and depression of the top and bottom of a hill are 60º and
30º respectively. Find the height of the hill
If the distance from the house to the hill is x, then
80/x = tan30°
x = 80√3
The height of the hill is thus
h = 80 + 80√3 tan60° = 320 m
Or, note that you have two similar 30-60-90° right triangles.
If the line-of-sight distance from the top of the house to the bottom of the hill is x, then
x/80 = 2
h/x = 2, so h = 2x = 320 m
To find the height of the hill, we can use trigonometry and create a right triangle.
Let's denote the height of the hill as "h".
First, let's consider the angle of elevation. From the top of the house, the angle of elevation to the top of the hill is 60 degrees. This forms a right triangle with the height of the hill (h) as the opposite side.
Using trigonometry, we know that the tangent of an angle is equal to the ratio of the opposite side to the adjacent side. In this case, the tangent of 60 degrees is equal to h (opposite side) divided by 80 meters (adjacent side).
So, we have the equation:
tan(60 degrees) = h / 80
Next, let's consider the angle of depression. From the top of the house, the angle of depression to the bottom of the hill is 30 degrees. This also forms a right triangle with the height of the hill (h) as the adjacent side.
Using trigonometry again, we know that the tangent of an angle is equal to the ratio of the opposite side to the adjacent side. In this case, the tangent of 30 degrees is equal to 80 meters (opposite side) divided by h (adjacent side).
So, we have the equation:
tan(30 degrees) = 80 / h
Now, we have a system of equations:
tan(60 degrees) = h / 80 (Equation 1)
tan(30 degrees) = 80 / h (Equation 2)
We can solve this system of equations to find the height of the hill.
First, let's solve Equation 1 for h:
h = 80 * tan(60 degrees)
Using a calculator, we find that tan(60 degrees) ≈ 1.732.
So, substituting this value into Equation 1:
h = 80 * 1.732
h ≈ 138.56 meters
Next, let's solve Equation 2 for h:
80 / h = tan(30 degrees)
Dividing both sides of the equation by tan(30 degrees):
h = 80 / tan(30 degrees)
Using a calculator, we find that tan(30 degrees) ≈ 0.577.
So, substituting this value into Equation 2:
h = 80 / 0.577
h ≈ 138.56 meters
Therefore, the height of the hill is approximately 138.56 meters.
To find the height of the hill, we can use trigonometry. Let's break down the problem step by step:
1. Draw a diagram: Draw a diagram with the house, the hill, and the angles of elevation and depression.
A (Top of the house)
------------------------
|\
| \
Hill | \ B
| \
| \
|____\
C (Bottom of the hill)
2. Identify the known values: We are given the height of the house (80m) and the angles of elevation and depression (60º and 30º).
3. Set up the trigonometric ratios: We will use the tangent function in this problem. The tangent of an angle is the ratio of the opposite side to the adjacent side.
Tan(A) = BC / AB (for angle A at the top of the house)
Tan(B) = AC / AB (for angle B at the bottom of the hill)
4. Solve for the unknowns: We can solve for the lengths BC and AC using the tangent ratios.
For angle A:
Tan(60º) = BC / AB
BC = AB * Tan(60º)
For angle B:
Tan(30º) = AC / AB
AC = AB * Tan(30º)
5. Calculate the height of the hill: The height of the hill is the sum of BC and AC.
Height of the hill = BC + AC
Height of the hill = AB * Tan(60º) + AB * Tan(30º)
We know that AB = height of the house (80m), so we substitute the value:
Height of the hill = 80 * Tan(60º) + 80 * Tan(30º)
6. Use a calculator to find the final answer: Evaluate the expression for the height of the hill using the tangent values of 60º and 30º:
Height of the hill ≈ 138.56 + 46.43
Height of the hill ≈ 184.99m
Therefore, the height of the hill is approximately 184.99 meters.