Chapter 4.3, Problem 77E

a.

To determine

Find the vertical rise of the inclined plane.

Answer to Problem 77E

  $\boxed{519.33feet}$

Explanation of Solution

Given information:

The Johnstown Inclined Plane in Pennsylvania is one of the longest and steepest hoists in the world. The r $35.4°$ railway cars travel a distance of  $\text{896}.\text{5}$ feet at an angle of approximately $35.4°$ , rising to a height of $1693.5$ feet above sea level.

Find the vertical rise of the inclined plane.

Calculation:

The figure of the Johnstown inclined Plane in Pennsylvania is given below. It is one of the longest and the steepest hoist in the world. The railway cars travel a distance of $\text{896}.\text{5}$ feet at an angle of $35.4°$ . rising to a height of $1693.5$ feet above the sea level.

We have to find the vertical rise of the inclined plane.

Let $x$ denote the vertical rise of the plane.

Now, from the above figure we can see that the length of the hypotenuse is $\text{896}.\text{5}$ feet.

The side opposite to the angle $35.4°$ is the vertical rise given by $x$ .

Now , we know that the trigonometric function which involves these two sides, that is the hypotenuse and the opposite to the angle is the sine function given by.

  $\sin \theta =\frac{\text{Opposite}}{\text{Hypotenuse}}$

Substituting the values in the above gven formula of sin, we have,

  $\sin 35.4°=\frac{x}{896.5}$

Or

  $x=896.5\times \sin 35.4°$

  $=896.5\times 0.579$

  $=519.33$

Hence, the vertical rise of the plane is $\boxed{519.33feet}$

b.

To determine

Find the elevation of the lower end of the inclined plane.

Answer to Problem 77E

  $\boxed{1693.5-519.33=1174.17feet}$

Explanation of Solution

Given information:

The Johnstown Inclined Plane in Pennsylvania is one of the longest and steepest hoists in the world. The r $35.4°$ railway cars travel a distance of  $\text{896}.\text{5}$ feet at an angle of approximately $35.4°$ , rising to a height of $1693.5$ feet above sea level.

Find the elevation of the lower end of the inclined plane.

Calculation:

The total height of the plane above the sea level is $1693.5$ feet.

From part a we see that the height of the plane is $519.33$ feet.

Hence, the elevation of the lower end of the inclined plane is

  $\boxed{1693.5-519.33=1174.17feet}$

c.

To determine

Find the rate at which they rise vertically.

Answer to Problem 77E

  $\boxed{173.7feet}$

Explanation of Solution

Given information:

The Johnstown Inclined Plane in Pennsylvania is one of the longest and steepest hoists in the world. The r $35.4°$ railway cars travel a distance of  $\text{896}.\text{5}$ feet at an angle of approximately $35.4°$ , rising to a height of $1693.5$ feet above sea level.

The cars move up the mountain at a rate of 300 feet per minute. Find the rate at which they rise vertically.

Calculation:

The cars move up the mountain at a rate $300$ feet per minute. We have to find the rate at which that rise vertically. We can assume to speed to be $y$ feet per minute. This can represent the side opposite to the angle $35.4°$ . Also the cars are currently moving up the mountain, which according to the figure above is giving by the hypotenuse.

Hence, the rate at which cars rise vertically is given by the following equation,

  $35.4°=\frac{y}{300}$

Or

  $y=300\times \sin 35.4°$

  $=896.5\times 0.579$

  $=173.7$

Hence, the rate at which cars rise vertically is $\boxed{173.7feet}$ .