Certainly! We can use the Law of Cosines to solve for the shortest length of the cable. Here’s how we can approach the problem using this method:

Given:
- The height of the mountain (\( h \)) = 3400 ft
- The angle of inclination (\( \theta \)) = 74°
- The horizontal distance from the base of the mountain to the point where the cable car starts (\( d \)) = 810 ft

First, let’s establish the components of the problem. We have a triangle with the following sides and angles:
- One side is the height of the mountain (3400 ft).
- One side is the horizontal distance from the point where the cable car starts to the base of the mountain (810 ft).
- The angle between these two sides is \( 74° + 90° = 164° \) because the angle of inclination of 74° with the horizontal means the angle with the vertical is 90° – 74° = 16°, and hence the angle in the triangle formed is \( 180° – 16° = 164° \).

Using the Law of Cosines, we can find the length of the cable \( C \) (the side opposite the 164° angle):

\[ C^2 = d^2 + h^2 - 2 \cdot d \cdot h \cdot \cos(164^\circ) \]

Substituting the given values:
\[ d = 810 \text{ ft} \]
\[ h = 3400 \text{ ft} \]
\[ \cos(164^\circ) \approx -0.9613 \]

So,
\[ C^2 = 810^2 + 3400^2 - 2 \cdot 810 \cdot 3400 \cdot (-0.9613) \]
\[ C^2 = 656100 + 11560000 + 5271960.4 \]
\[ C^2 = 17828160.4 \]
\[ C = \sqrt{17828160.4} \approx 4223.53 \text{ ft} \]

Thus, the shortest length of cable needed is approximately 4223.53 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. How does the Law of Cosines compare to using the Pythagorean theorem in this scenario?
2. What are some other applications of the Law of Cosines?
3. How would the problem change if the angle of inclination was less than 74°?
4. Can you solve this problem using a vector approach?
5. How do changes in the horizontal distance affect the required cable length?
6. What other trigonometric laws are useful for solving real-world problems?
7. How would you solve for the length of the cable if the mountain height changed?
8. What practical considerations might affect the installation of the cable car?

**Tip:** The Law of Cosines is especially useful for solving triangles that are not right-angled, as it relates all three sides of a triangle to one of its angles.

Learn how to use the Law of Cosines to calculate the shortest length of a cable needed for a mountain cable car. Given specific dimensions and angles, find out the step-by-step solution and understand related trigonometric concepts. Suitable for advanced high school students and beyond.

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