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 the earth.
a. Solve the equation for r. Solve the equation for C. Solve the equation for w.
b. Suppose that an object is 350 pounds when it is at sea level. Find the value of C that makes the equation true. (Sea level is approximately 4,000 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
i. 1,100 feet below sea level (for example, near the bottom of the ocean)
ii. 21,500 feet above sea level (for example, on the top of a high mountain)
the equation is not given..
a. To solve the equation for r, C, and w, we can start with the given equation:
w = C / r
To solve for r, we can rearrange the equation as follows:
w * r = C
Dividing both sides by w, we find:
r = C / w
To solve for C, we can rearrange the equation as follows:
w = C / r
Multiplying both sides by r, we get:
w * r = C
So, C = w * r
To solve for w, we can rearrange the equation as follows:
w = C / r
b. We are given that the object weighs 350 pounds when it is at sea level (4,000 miles from the center of the Earth). Let's substitute the given values into the equation to solve for C:
w = C / r
350 = C / 4,000
To solve for C, we can multiply both sides by 4,000:
350 * 4,000 = C
C = 1,400,000
Therefore, the value of C that makes the equation true is 1,400,000.
c. Now, using the value of C we found in the previous question, let's determine how much the object would weigh at different elevations:
i. 1,100 feet below sea level:
To find the weight, we need to determine the distance from the center of the Earth. Since we are below sea level, the distance would be the distance from sea level (4,000 miles) minus the depth (1,100 feet).
Convert 1,100 feet to miles by dividing by 5,280 (1 mile = 5,280 feet):
1,100 ft / 5,280 ft/mile = 0.2083 miles
r = 4,000 miles - 0.2083 miles = 3,999.7917 miles
Now, substitute the values into the equation:
w = C / r
w = 1,400,000 / 3,999.7917
w ≈ 350.0002 pounds
Therefore, the object would weigh approximately 350.0002 pounds.
ii. 21,500 feet above sea level:
To find the weight, we need to determine the distance from the center of the Earth. Since we are above sea level, the distance would be the distance from sea level (4,000 miles) plus the height (21,500 feet).
Convert 21,500 feet to miles by dividing by 5,280 (1 mile = 5,280 feet):
21,500 ft / 5,280 ft/mile = 4.0759 miles
r = 4,000 miles + 4.0759 miles = 4,004.0759 miles
Now, substitute the values into the equation:
w = C / r
w = 1,400,000 / 4,004.0759
w ≈ 349.8553 pounds
Therefore, the object would weigh approximately 349.8553 pounds.