A steep mountain is inclined 74 degree to the horizontal and rises to a height of 3400 ft above the surrounding plain. A cable car is to be installed running to the top of the mountain from a point 810 ft out in the plain from the base of the mountain. Find the shortest length of cable needed.

To find the shortest length of cable needed for the cable car, we can use trigonometry and the Pythagorean theorem. Let's break down the problem step by step.

1. **Determine the length of the incline (hypotenuse) from the base to the top of the mountain.**

Given:
- Height of the mountain, \( h = 3400 \) ft
- Incline angle, \( \theta = 74^\circ \)

The length of the incline (let's call it \( L \)) can be found using the sine function:
\[ \sin(74^\circ) = \frac{h}{L} \]
\[ L = \frac{h}{\sin(74^\circ)} = \frac{3400}{\sin(74^\circ)} \]

Let's calculate this:
\[ \sin(74^\circ) \approx 0.9613 \]
\[ L = \frac{3400}{0.9613} \approx 3537.15 \text{ ft} \]

2. **Determine the horizontal distance from the base of the mountain to the point where the cable car starts.**

Given:
- Distance from the base of the mountain to the point where the cable car starts, \( d = 810 \) ft

3. **Calculate the horizontal component of the incline.**

The horizontal component \( x \) of the incline can be found using the cosine function:
\[ \cos(74^\circ) = \frac{x}{L} \]
\[ x = L \cdot \cos(74^\circ) \]

Let's calculate this:
\[ \cos(74^\circ) \approx 0.2756 \]
\[ x = 3537.15 \cdot 0.2756 \approx 975.15 \text{ ft} \]

4. **Determine the horizontal distance from the point where the cable car starts to the top of the mountain.**

The total horizontal distance from the point where the cable car starts to the top of the mountain is:
\[ d + x = 810 + 975.15 = 1785.15 \text{ ft} \]

5. **Calculate the shortest length of the cable using the Pythagorean theorem.**

We have a right triangle where:
- One leg is the total horizontal distance \( 1785.15 \) ft
- The other leg is the height of the mountain \( 3400 \) ft

The shortest length of the cable (hypotenuse) \( C \) is:
\[ C = \sqrt{(1785.15)^2 + (3400)^2} \]

Let's calculate this:
\[ C = \sqrt{(1785.15)^2 + (3400)^2} \approx \sqrt{3182387.32 + 11560000} \approx \sqrt{14742387.32} \approx 3839.14 \text{ ft} \]

So, the shortest length of cable needed is approximately \( 3839.14 \) ft.

Would you like further details on any specific step, or do you have any questions?

Here are some related questions for further exploration:
1. What is the exact length of the cable if the incline angle were different?
2. How would the required length change if the height of the mountain increased?
3. What trigonometric identities are used in solving problems involving inclines?
4. How does the Pythagorean theorem apply to right triangles?
5. Can you derive the formula for the length of the incline using only trigonometric identities?
6. What is the significance of understanding horizontal and vertical components in physics?
7. How would the length of the cable change if the starting point in the plain was farther out?
8. What are the potential real-world applications of these trigonometric principles?

**Tip:** When solving trigonometric problems, always ensure your calculator is set to the correct mode (degrees or radians) as per the given angles.

Learn how to calculate the shortest length of cable needed for a cable car inclined at 74 degrees to a mountain rising 3400 ft above a plain. Includes step-by-step trigonometric calculations using the sine function, cosine function, and the Pythagorean theorem. Suitable for advanced high school students.

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