Questions: A steep mountain is inclined 79° to the horizontal and rises 3100 ft above the surrounding plain. A cable car is to be installed from a point 900 ft from the base to the top of the mountain, as shown. Find the shortest length of cable needed. Please round the answer to the nearest integer, ft

Solution Steps

The problem provides the following information:

• The mountain is inclined at an angle of 79° to the horizontal.
• The height of the mountain is 3100 ft.
• The horizontal distance from the base of the mountain to the point where the cable car is to be installed is 900 ft.

We need to find the hypotenuse of the right triangle formed by the height of the mountain, the horizontal distance, and the cable length. The height of the mountain (3100 ft) is the opposite side, and the horizontal distance (900 ft) is the adjacent side.

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b): \[ c^2 = a^2 + b^2 \] Here, \( a = 3100 \) ft and \( b = 900 \) ft.

\[ c^2 = 3100^2 + 900^2 \] \[ c^2 = 9610000 + 810000 \] \[ c^2 = 10430000 \] \[ c = \sqrt{10430000} \] \[ c \approx 3229.48 \]

Final Answer