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. (Round your answers to two decimal places.)
look at the picture in the link below:
www.webassign.net/larpcalclim2/4-3-088.gif
(a) Find the vertical rise of the inclined plane.
(b) Find the elevation of the lower end of the inclined plane.
(c) The cars move up the mountain at a rate of 300 feet per minute. Find the rate at which they rise vertically.
To solve these questions, we can use trigonometry. Let's look at each part:
(a) To find the vertical rise of the inclined plane, we need to calculate the length of the opposite side of the triangle in the picture.
In this case, the opposite side represents the vertical rise. Using the given information, we know the hypotenuse (the distance traveled) is 896.5 feet and the angle is 35.4°.
We can use the trigonometric function sine (sin) to find the opposite side. Remember that sine is the ratio of the opposite side to the hypotenuse.
So, to find the vertical rise:
Vertical rise = Hypotenuse * sine(angle)
Vertical rise = 896.5 * sin(35.4°)
Calculating this value will give us the answer for part (a).
(b) To find the elevation of the lower end of the inclined plane, we need to calculate the length of the adjacent side of the triangle in the picture.
In this case, the adjacent side represents the elevation of the lower end. Again, using the given information, we know the hypotenuse (the distance traveled) is 896.5 feet and the angle is 35.4°.
We can use the trigonometric function cosine (cos) to find the adjacent side. Remember that cosine is the ratio of the adjacent side to the hypotenuse.
So, to find the elevation of the lower end:
Elevation = Hypotenuse * cosine(angle)
Elevation = 896.5 * cos(35.4°)
Calculating this value will give us the answer for part (b).
(c) To find the rate at which the cars rise vertically, we can use the given information about the rate at which the cars move up the mountain.
The rate at which they rise vertically will be the same as the vertical component of their velocity.
Given that the cars move up the mountain at a rate of 300 feet per minute, the rate at which they rise vertically will be the same.
Therefore, the rate at which they rise vertically is 300 feet per minute.
By following these steps, we can find the answers to all three parts of this question.