What is the shortest length of the cable?

Question

A steep mountain is inclined 64° from the horizontal and rises to a height of 3150 ft above the surrounding plain. A cable car is to be installed running from a point 890 ft from the base of the mountain to the top of the mountain .The shortest length of cable   ft.   

Answer 1

Sketch a diagram of a right triangle with an angle of 64° and a height of 3150 ft. Label the base of the triangle (the base of the mountain) x.

Now we can say: tan64º = 3150/x

x = 1536.36 ft

Draw another line from the top of the mountain to a point 890 ft beyond x. You have another right triangle with a height of 3150 ft and a base of 1536.36 + 890 = 2426.36 ft.

The shortest length of cable will be the hypotenuse of this new triangle. Use Pythagoras' Theorem to calculate it's length.