Sandra wants to find the height of a moutain. From her first location on the ground, she finds the angle of elevation to the top of the mountain to be 35 degrees 15 minutes. After moving 1000 meters closer to the moutain on level ground, she finds the elevation to be 50 degrees and 25 minutes . Find the height of the moutain to the nearest foot

make a sketch labeling the height of the mountain as PQ, with P as the top of the mountain.

Label the 2 points of observation as A and B, with B the closer point,
In triang ABP,
angle A = 35°15'
angle ABP = 180 - 50°25' = 129°35'
then angle APB = 15°10'
and AB = 1000 m

by the sine law:
BP/sin35°15' = 1000/sin15°10'
BP = 2205.9788..

now in the right-angled triangle BPQ
PQ/PB = sin50°25'
PQ = Height of mountain = 1700.145 m
or appr 1700 m

Did you want the answer in feet?
Why did you have metric units ?
I will let you do the conversion, if needed.

To find the height of the mountain, we can use trigonometry. Let's denote the height of the mountain as "h" and the distance from Sandra's first location on the ground to the mountain as "x".

Step 1: Convert the angles from degrees and minutes to decimal form.
- The first angle of elevation, 35 degrees 15 minutes, can be converted to decimal form as follows:
35 degrees + (15 minutes / 60 minutes) = 35.25 degrees

- The second angle of elevation, 50 degrees 25 minutes, can be converted to decimal form as follows:
50 degrees + (25 minutes / 60 minutes) = 50.4167 degrees (rounded to 4 decimal places)

Step 2: Use trigonometry to set up equations based on the angles of elevation.

- From Sandra's first location, when the angle of elevation is 35.25 degrees, we have:
tan(35.25 degrees) = h / x (Equation 1)

- After moving 1000 meters closer to the mountain, the distance from Sandra's second location to the mountain is (x - 1000).
When the angle of elevation is 50.4167 degrees, we have:
tan(50.4167 degrees) = h / (x - 1000) (Equation 2)

Step 3: Solve the equations simultaneously to find the value of "h".

- Rearrange Equation 1 to solve for h:
h = x * tan(35.25 degrees)

- Substitute this value into Equation 2:
tan(50.4167 degrees) = (x * tan(35.25 degrees)) / (x - 1000)

- Multiply both sides by (x - 1000):
(x - 1000) * tan(50.4167 degrees) = x * tan(35.25 degrees)

- Expand and rearrange the equation:
x * tan(50.4167 degrees) - (1000 * tan(50.4167 degrees)) = x * tan(35.25 degrees)

- Simplify further:
x * (tan(50.4167 degrees) - tan(35.25 degrees)) = 1000 * tan(50.4167 degrees)

- Finally, solve for x:
x = (1000 * tan(50.4167 degrees)) / (tan(50.4167 degrees) - tan(35.25 degrees))

Step 4: Substitute the value of x back into Equation 1 to find h.

- h = x * tan(35.25 degrees)

- Substitute the value of x into Equation 1 to find h.

Step 5: Round the value of h to the nearest foot to get the final answer.

- Round the value of h to the nearest foot.

To find the height of the mountain, we can use trigonometry and the knowledge of angles of elevation.

First, let's label the triangle formed by Sandra's locations and the top of the mountain:

x (distance between the first location and the mountain top)
___________
|\
| \
h | \
(height) | \
| \
|θ \(angle of elevation)
|______\
y (distance between the second location and the mountain top)

From the information given, we have two right triangles:

1. First Triangle:
- Angle of elevation: 35 degrees 15 minutes
- Opposite side (h): height of the mountain
- Adjacent side (x): distance between the first location and the mountain top

2. Second Triangle:
- Angle of elevation: 50 degrees 25 minutes
- Opposite side (h): height of the mountain
- Adjacent side (y): distance between the second location and the mountain top

To solve this problem, we need to find the value of 'h' (height of the mountain). Here's how we can do it:

1. Convert the angles from degrees and minutes to decimal form:
- First angle: 35 degrees 15 minutes = 35 + 15/60 = 35.25 degrees
- Second angle: 50 degrees 25 minutes = 50 + 25/60 = 50.4167 degrees

2. Use the tangent function to find the height of the mountain in each triangle:
- In the first triangle: tan(35.25 degrees) = h / x
- In the second triangle: tan(50.4167 degrees) = h / y

3. Solve the system of equations for 'h':
- Divide the second equation by the first equation to eliminate 'h':
(tan(50.4167 degrees) / tan(35.25 degrees)) = (h / y) / (h / x)
- Simplify the equation to solve for 'h':
h = (y * tan(35.25 degrees)) / tan(50.4167 degrees)

4. Plug in the values given:
- Distance moved (y) = 1000 meters
- Calculate h using the equation from step 3.

Once you have the value of 'h' (height of the mountain), you can convert it to feet (or any other desired unit) and round it to the nearest foot.

Note: Make sure to use a scientific calculator or trigonometric calculator to evaluate the trigonometric functions accurately.