You came across an egg that was a perfect sphere. You placed it in a tub of water and found that it displaced exactly 280,000 cubic centimetres. However, it was floating on the water, and only 72% of the surface area of the egg was below the water line.
What was the total volume of the egg, in cubic centimetres?
Isn't the volume of a sphere 3/4 pi r^3?
How would I go about solving this problem?
First of all the volume of a sphere is 4/3 pi r^3, not 3/4 pi r^3.
You have to determine how much of the volume of the egg was submerged, knowing what fraction of the area was submerged. They are not the same fractions. The solution requires calculus. Let y be the distance of the center of the sphere below the water line. The area above the water line is 28% of 4 pi r^2 or 1.12 pi r^2
Derive a formula for the angle theta between the vertical and the water line, which is cos^-1 (y/r), using calculus, and solve for theta.
I think you will find that the area above water is 2 pi r^2 cos theta. Since that is 1.12 pi r^2, the angle theta is cos^-1 0.56 = 55.94 degrees.
Once you know theta, derive another equation for the fraction of the sphere below the water line, also in terms of theta. Then you can solve for the total sphere volume.
To solve this problem, we need to use the information given to find the total volume of the egg. Let's break down the steps:
Step 1: Calculate the total volume of the egg using the formula for the volume of a sphere, which is V = (4/3) * pi * r^3, where V is the volume and r is the radius.
But in this case, we don't have the radius of the sphere, so we need to find it using the information about displacement and surface area below the water line.
Step 2: Calculate the volume of water displaced by the egg. We know that it displaced exactly 280,000 cubic centimeters.
Step 3: Use the volume of water displaced to find the volume of the sphere submerged in the water.
Step 4: Calculate the surface area of the egg using the information that only 72% of it is below the water line.
Step 5: Determine the radius of the sphere using the surface area below the water line.
Step 6: Plug the calculated radius into the formula for the volume of a sphere to find the total volume of the egg.
Now let's go through each step in detail:
Step 1: Calculate the total volume of the egg.
V = (4/3) * pi * r^3
Step 2: Calculate the volume of water displaced by the egg.
Given: Displacement = 280,000 cubic centimeters
Step 3: Use the volume of water displaced to find the volume of the sphere submerged in the water.
The volume of the submerged sphere is equal to the volume of water displaced.
Step 4: Calculate the surface area of the egg.
Given: Only 72% of the surface area of the egg is below the water line.
Step 5: Determine the radius of the sphere.
To find the radius, we need to use the surface area below the water line.
Step 6: Plug the calculated radius into the formula for the volume of a sphere to find the total volume of the egg.
V = (4/3) * pi * r^3
Now, let's go ahead and calculate each step to find the total volume of the egg.