Answer the following questions. Use Equation Editor to write mathematical expressions and equations. First, save this file to your hard drive by selecting Save As from the File menu. Click the white space below each question to maintain proper formatting.
Hint: Pay attention to the units of measure. You may have to convert from feet to miles several times in this assignment. You can use 1 mile = 5,280 feet for your conversions.
1. Many people know that the weight of an object varies on different planets, but did you know that the weight of an object on Earth also varies according to the elevation of the object? In particular, the weight of an object follows this equation: , where C is a constant, and r is the distance that the object is from the center of Earth.
a. Solve the equation for r.
b. Suppose that an object is 100 pounds when it is at sea level. Find the value of C that makes the equation true. (Sea level is 3,963 miles from the center of the Earth.)
c. Use the value of C you found in the previous question to determine how much the object would weigh in
i. Death Valley (282 feet below sea level).
ii. the top of Mount McKinley (20,320 feet above sea level).
2. The equation gives the distance, D, in miles that a person can see to the horizon from a height, h, in feet.
a. Solve this equation for h.
b. Long’s Peak in Rocky Mountain National Park, is 14,255 feet in elevation. How far can you see to the horizon from the top of Long’s Peak? Can you see Cheyenne, Wyoming (about 89 miles away)? Explain your answer.
How would you like us to help you with this assignment?
I can not see your equation but assume it is of the form
W = C/r^2
then
r^2 = C/W
r = sqrt (C/W)
Now sea level in feet is
r = 3963 miles (5280 ft/mile) = 20.9 * 10^6 feet
so
100 = C/(20.9*10^6)^2
C = 4.37*10*16
In death valley r = 20.9*10^6 - 282
r = 20.899*10^6
r^2 = 436.8*10*12
W = 100.05 tiny bit heavier
do the same thing on the mountain. It will be slightly lighter
For the second problem I also can not see your formula
It will be of form
D =sqrt (h(2R+h))
if h <<r (which it is here)
D = sqrt (2 R h)
then
D^2 = 2 R h
h = D^2/(2R)
Now
see how far you can see the horizon from Long's peak (find D for h = height of Long's peak)
then see how far you can see from Cheyenne (You will have to look up how high the city is)
add those two to find out if it is more than 89 miles.
We will have to assume that nothing sticks up between them.
1. The weight of an object on Earth varies according to the elevation of the object. The weight of an object is given by the equation:
W = C / (r^2)
a. To solve the equation for r, we need to isolate r.
Start by multiplying both sides of the equation by r^2:
W * r^2 = C
Then divide both sides of the equation by the weight W:
r^2 = C / W
Finally, take the square root of both sides of the equation to solve for r:
r = sqrt(C / W)
b. Suppose an object weighs 100 pounds when it is at sea level, which is 3,963 miles from the center of the Earth. We can use this information to find the value of C that makes the equation true.
Substituting the given values into the equation:
100 = C / (3963)^2
Simplifying the equation:
100 = C / 15705369
To solve for C, multiply both sides of the equation by 15705369:
C = 100 * 15705369
C ≈ 1,570,536,900
Therefore, the value of C that makes the equation true is approximately 1,570,536,900.
c. To determine how much the object would weigh at different elevations, we can use the value of C found in the previous question and the equation:
W = C / (r^2)
i. Death Valley is 282 feet below sea level. To find the weight of the object in Death Valley, we need to calculate the value of r in miles by converting the height in feet to miles:
r = 282 feet / 5280 feet/mile ≈ 0.053 miles
Substituting the values into the equation:
W = 1,570,536,900 / (0.053)^2
Calculating the weight:
W ≈ 1,570,536,900 / 0.002809 ≈ 559,545,274.8 pounds
Therefore, the object would weigh approximately 559,545,274.8 pounds in Death Valley.
ii. The top of Mount McKinley is 20,320 feet above sea level. To find the weight of the object at the top of Mount McKinley, we need to calculate the value of r in miles:
r = 20,320 feet / 5280 feet/mile ≈ 3.85 miles
Substituting the values into the equation:
W = 1,570,536,900 / (3.85)^2
Calculating the weight:
W ≈ 1,570,536,900 / 14.8225 ≈ 105,877,209.5 pounds
Therefore, the object would weigh approximately 105,877,209.5 pounds at the top of Mount McKinley.
2. The distance, D, in miles that a person can see to the horizon from a height, h, in feet is given by the equation:
D = sqrt(1.5 * h)
a. To solve this equation for h, we need to isolate h. Start by squaring both sides of the equation:
D^2 = (1.5 * h)
Then divide both sides of the equation by 1.5:
h = D^2 / 1.5
b. Long's Peak in Rocky Mountain National Park has an elevation of 14,255 feet. To find how far you can see to the horizon from the top of Long's Peak, we can use the equation:
D = sqrt(1.5 * h)
Substituting the given value of h:
D = sqrt(1.5 * 14255)
Calculating the distance:
D ≈ sqrt(21382.5) ≈ 146.16 miles
Therefore, you can see approximately 146.16 miles to the horizon from the top of Long's Peak.
As for whether you can see Cheyenne, Wyoming (about 89 miles away), the distance to the horizon is greater than 89 miles. Therefore, you should be able to see Cheyenne, Wyoming from the top of Long's Peak.