When you look out over an unobstructed landscape or seascape, the distance to the visible horizon depends on your height above the ground. The equation d=3/2h is a good estimate of this, in which d=distance to horizon in miles and h=height of viewer about the ground in feet. It is possible to find different distances to the horizon given different elevations. (a)solve the above equation for the height h, (b)Can you see a object that is 100 miles away from the top of the Empire State Building? (c)Long's Peak in Rocky Mountain National Park, is 14,255 feet in elevation. How far can you see the horizon from the top of Long's Peak? Can you see Cheyenne, Wyoming (about 89 miles away)? Explain your answer.
(a) To solve the equation d = (3/2)h for the height h, we can rearrange the equation by isolating h on one side.
Starting with the equation:
d = (3/2)h
We can multiply both sides of the equation by (2/3) to cancel out the coefficient of h:
(d) * (2/3) = (3/2) * (2/3) * h
This simplifies to:
(2d/3) = h
So the equation solved for h becomes:
h = (2d/3)
(b) To determine if you can see an object that is 100 miles away from the top of the Empire State Building, we can substitute the values into the equation and compare it to the height of the building.
Given:
d = 100 miles
h = height of the Empire State Building
Using the equation h = (2d/3) and substituting d = 100 miles, we can solve for h:
h = (2 * 100 miles) / 3
h = 200 miles / 3
Now, let's compare the height of the Empire State Building to the calculated value of h. If the height of the Empire State Building is greater than or equal to h, then you should be able to see an object that is 100 miles away. Otherwise, you would not be able to see it.
(c) For Long's Peak, which is at an elevation of 14,255 feet, we can use the equation d = (3/2)h to determine the distance to the horizon.
Given:
h = 14,255 feet
Using the equation d = (3/2)h, we can substitute the given value of h and solve for d:
d = (3/2) * 14,255 feet
d = 21,382.5 feet
To convert feet to miles, divide the distance by 5,280 (since there are 5,280 feet in a mile):
d = 21,382.5 feet / 5,280 feet per mile
d ≈ 4.05 miles
Therefore, from the top of Long's Peak, you should be able to see the horizon at a distance of approximately 4.05 miles.
As for seeing Cheyenne, Wyoming, which is about 89 miles away, you would not be able to see it from the top of Long's Peak. The calculated distance to the horizon from Long's Peak is only about 4.05 miles, which is significantly shorter than the distance to Cheyenne.