A woman measures the angle of elevation from her current location to a mountaintop to be 12.0°. After walking 1.10 km closer to the mountain on level ground, she measures the angle again and finds it to be 14.0°. What is the height of the mountain?
1. Draw a hor. line.
2. Draw a ver. line to the right-end of the hor. line and label it h for height.
3. From the left-end of the hor. line, draw a hypotenuse to the top of the ver. line. The angle it makes with the
hor. is 12o.
4. Draw a 2nd hyp. to the right of the first and label the distance between them X1 = 1.1 km. The remaining section
is X2.
Tan12 = h/(x1+x2) = h/(1.1+x2)
h = (1.1+x2)*Tan12
Tan14 = h/x2
h = x2*Tan14
x2*Tan14 = (1.1+x2)*Tan12
0.249x2 = 0.213x2 + 0.234
0.249x2-0.213x2 = 0.234
0.036x2 = 0.234
X2 = 6.5 km
h = X2*Tan14 = 6.5 * Tan14 = 1.62 km.
To solve this problem, we can use the tangent function.
Let's assume the distance from the woman's initial location to the mountain is 'x' kilometers.
Using the first measurement, we have:
Tan(12°) = height of the mountain / x
Using the second measurement, we have:
Tan(14°) = height of the mountain / (x - 1.10)
Now we can set up an equation and solve for the height of the mountain:
Tan(12°) = Tan(14°) * (x / (x - 1.10))
First, let's find the values of Tan(12°) and Tan(14°):
Tan(12°) ≈ 0.2126
Tan(14°) ≈ 0.2493
Now, we can solve the equation:
0.2126 = 0.2493 * (x / (x - 1.10))
Cross-multiplying gives:
0.2126 * (x - 1.10) = 0.2493 * x
0.2126x - 0.2349 = 0.2493x
0.0367x = 0.2349
x = 0.2349 / 0.0367
x ≈ 6.4033
So, the distance from the woman's initial location to the mountain is approximately 6.4033 kilometers.
Now, we can calculate the height of the mountain using the first measurement:
Tan(12°) = height of the mountain / x
0.2126 = height of the mountain / 6.4033
height of the mountain ≈ 0.2126 * 6.4033
height of the mountain ≈ 1.3586 kilometers
Therefore, the height of the mountain is approximately 1.3586 kilometers.
To find the height of the mountain, we can use trigonometry. We can use the tangent function to relate the angle of elevation to the height and the distance.
Let's denote the height of the mountain as "h" and the distance from the woman's starting point to the mountain as "x". Given that the woman initially measures the angle of elevation to be 12.0 degrees, we can set up the following equation:
tan(12.0°) = h / x (Equation 1)
After the woman walks 1.10 km closer to the mountain, the distance from her new location to the mountain would be "x - 1.10 km". And she measures the new angle of elevation to be 14.0 degrees. We can set up another equation using the new information:
tan(14.0°) = h / (x - 1.10 km) (Equation 2)
Now we have a system of two equations (Equation 1 and Equation 2) that we can solve to find the height of the mountain.
To find the value of "h", we can isolate it in Equation 1:
h = x * tan(12.0°) (Equation 3)
Substituting Equation 3 into Equation 2, we can eliminate the "h" variable and solve for "x":
tan(14.0°) = (x * tan(12.0°)) / (x - 1.10 km)
To solve this equation, we can multiply both sides by (x - 1.10 km):
(x - 1.10 km) * tan(14.0°) = x * tan(12.0°)
Expanding the equation and simplifying:
xtan(14.0°) - 1.10 km * tan(14.0°) = x * tan(12.0°)
Now, isolate the "x" term:
xtan(14.0°) - x * tan(12.0°) = 1.10 km * tan(14.0°)
x * (tan(14.0°) - tan(12.0°)) = 1.10 km * tan(14.0°)
Finally, solve for "x":
x = (1.10 km * tan(14.0°)) / (tan(14.0°) - tan(12.0°))
Once we find the value of "x", we can substitute it back into Equation 1 to calculate the height "h":
h = x * tan(12.0°)
This will give us the height of the mountain.