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.
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 Mt McKinley (20,320 feet above sea level)
What's the question?
a. To solve the equation for r, we need to isolate r on one side of the equation. The equation is given as:
Weight = C / r
To isolate r, we can start by multiplying both sides of the equation by r:
Weight * r = C
Next, divide both sides of the equation by Weight:
r = C / Weight
So, the equation for r is r = C / Weight.
b. We are given that the object weighs 100 pounds at sea level, which is 3,963 miles from the center of the earth. Using this information, we can substitute the values into the equation and solve for C:
Weight = C / r
100 = C / 3963
To solve for C, multiply both sides of the equation by 3963:
C = 100 * 3963
C = 396,300
Therefore, the value of C that makes the equation true is 396,300.
c. Now, using the value of C we found in the previous question, we can determine how much the object would weigh in the given locations:
i. Death Valley (282 feet below sea level):
To calculate the distance of Death Valley from the center of the earth, we subtract 282 feet from the distance at sea level (3,963 miles). We need to convert 282 feet to miles by dividing it by the number of feet in a mile (5280):
Distance = 3,963 - (282 / 5280)
Weight = C / Distance
Plug in the values:
Weight = 396,300 / (3,963 - 0.053))
Simplify:
Weight ≈ 101.017 pounds
Therefore, the object would weigh approximately 101.017 pounds in Death Valley.
ii. The top of Mt McKinley (20,320 feet above sea level):
To calculate the distance of the top of Mt McKinley from the center of the earth, we add 20,320 feet to the distance at sea level:
Distance = 3,963 + (20,320 / 5280)
Weight = C / Distance
Plug in the values:
Weight = 396,300 / (3,963 + 3.853))
Simplify:
Weight ≈ 98.084 pounds
Therefore, the object would weigh approximately 98.084 pounds at the top of Mt McKinley.