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.)
www.webassign.net/larpcalclim2/4-3-088.gif
(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) h = 896.5sin35.4 = 519.33 ft
b) 1693.5 - 519.33 = 1174.17 ft
c) 300sin35.4 = 173.80 ft/min
is this correct??
The height is found via
h/896.5 = sin 35.4°
h = 519.3 ft
The lower end's elevation is just 1693.5-h, right?
hint: feet/min means divide rise by time.
I agree with all of those. Make sure you know how to work them out in case you have other questions like this.
hey oobleck, thanks, is my answers correct?
looks good to me
To find the answers to the given questions, we can use trigonometric functions like sine, cosine, and tangent.
(a) To find the vertical rise of the inclined plane, we can use the sine function. The formula is:
Vertical Rise = Distance Traveled * sin(Angle of Incline)
In this case, the Distance Traveled is 896.5 feet, and the Angle of Incline is 35.4 degrees.
Vertical Rise = 896.5 * sin(35.4°)
Calculating this value will give us the answer to part (a).
(b) To find the elevation of the lower end of the inclined plane, we need to subtract the Vertical Rise from the starting elevation. The starting elevation is given as 1693.5 feet above sea level.
Elevation of lower end = Starting elevation - Vertical Rise
Calculating this value will give us the answer to part (b).
(c) To find the rate at which the cars rise vertically, we can use the tangent function. The formula is:
Rate of vertical ascent = Rate of horizontal motion / cos(Angle of Incline)
In this case, the Rate of horizontal motion is given as 300 feet per minute, and the Angle of Incline is 35.4 degrees.
Rate of vertical ascent = 300 / cos(35.4°)
Calculating this value will give us the answer to part (c).