A steep mountain is inclined 74° to the horizontal and rises 3,100 ft above the surrounding plain. A cable car is to be installed from a point 850 ft from the base and attached to the top of the mountain, as shown in the following figure.
8503,100
A mountain is shown. The moutain is 3100 ft tall and makes an angle of 74° with the horizon. A cable car line begins 850 ft from the base of the mountain and goes to the top of the mountain.
Find the shortest length (in ft) of cable needed

We can solve this problem by using trigonometry. To find the shortest length of cable needed, we need to determine the distance from the point 850 ft away from the base of the mountain to the top of the mountain. This distance is along the hypotenuse of the right triangle formed by:

- The horizontal distance from the point where the cable car starts (850 ft),
- The vertical height of the mountain (3,100 ft),
- The angle of elevation of the mountain with respect to the horizontal (74°).

### Step 1: Define the known variables
- Height of the mountain \( h = 3,100 \, \text{ft} \)
- Angle of inclination \( \theta = 74^\circ \)
- Horizontal distance \( d = 850 \, \text{ft} \)

### Step 2: Use the Pythagorean theorem
Since the height of the mountain and the horizontal distance form the two legs of a right triangle, we can find the shortest length of cable needed, which is the hypotenuse of the triangle.

The hypotenuse \( L \) is calculated as:
\[
L = \sqrt{h^2 + d^2}
\]

### Step 3: Substitute the known values
\[
L = \sqrt{3,100^2 + 850^2}
\]

Now, let's calculate the length.The shortest length of cable needed is approximately 3,214.42 feet.

Would you like further clarification, or do you have any other questions?

Here are some related questions to explore:
1. How would the length of the cable change if the height of the mountain were different?
2. What if the horizontal distance from the base was increased to 1,000 feet—how would that affect the cable length?
3. How could the Pythagorean theorem be applied in real-life engineering situations like this one?
4. What would happen if the angle of inclination of the mountain were less than 74°?
5. Could trigonometric functions other than the Pythagorean theorem be used to solve this problem?

**Tip:** In right-angled triangles, the Pythagorean theorem is a quick way to find unknown sides, especially when two sides are known.

A steep mountain is inclined 74° to the horizontal and rises 3,100 ft above the surrounding plain. A cable car is to be installed from a point 850 ft from the base and attached to the top of the mountain. Find the shortest length (in ft) of cable needed.

Learn how to determine the shortest length of cable needed to connect a point 850 ft from the base of a mountain to its top using trigonometry. This problem involves applying the Pythagorean theorem to a right triangle formed by the mountain's height of 3,100 ft and an inclination angle of 74°. Suitable for high school students in Grades 9-12.

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