The Johnstown Inclined Plane in Pennsylvania is one of the longest and steepest hoists in the world. The railway cars travel a distance of 896.5 feet at an angle of approximately 35.4°, rising to a height of 1693.5 feet above sea level. (see figure in the website). (Round your answers to two decimal places.)

(a) Find the vertical rise of the inclined plane. (in feet)

(b) Find the elevation of the lower end of the inclined plane. (in feet)

(c) The cars move up the mountain at a rate of 300 feet per minute. Find the rate at which they rise vertically. (in ft/min)

a. 519.35 ft b. 1174.15 ft c. 173.78 ft/min

Explanation:

a. Since the length of the incline = 896.54 ft, and the angle of the incline = 35.4°. Let h be the vertical rise of the incline. From trigonometric identities,

sin35.4° = h/896.54 ⇒ h = 896.54sin35.4° = 519.35 ft

b. Since the top of the incline is 1693,5 ft above sea level, and the vertical height of the incline is 519.35 ft, the elevation of the lower end of the inclined plane is 1693.5ft - 519.35ft = 1174.15 ft

c. The cars move up the mountain at a rate of 300 ft/min along the inclined plane. The rate at which they rise vertically is the vertical component of the velocity = 300sin35.4° ft/min = 173.78 ft/min