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.

Question

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?

Answer 1

To solve this problem, we can use the concept of displacement and the volume formula for a sphere. Let's begin by applying the volume formula for a sphere, which is indeed (4/3)πr^3.

Let's assign some variables to the information given in the problem:
- V is the total volume of the egg (what we want to find).
- R is the radius of the egg.
- D is the displacement of the egg in the water.

The volume of the egg can be represented as V = (4/3)πR^3, where R is the radius of the sphere.

Now, let's incorporate the given information. The egg displaces 280,000 cubic centimeters of water, which means that the volume of the egg is equal to the volume of the water it displaces. Therefore, we have:

V = D

However, we still need to find the value of D. We know that only 72% of the surface area of the egg is below the water line. This means that 28% of the surface area is above the water. Since the egg is floating, the weight of the water displaced by the submerged part is equal to the weight of the egg. This is due to the principle of buoyancy.

Using this information, we can establish the relationship between the surface area of the egg and the volume of water displaced. We have:

V = D = (total surface area of the egg) × (depth of the submerged part in water)

The total surface area of a sphere can be derived using the formula 4πR^2. We need to consider that only 72% of the total surface area is submerged, so the surface area submerged in water is:

(0.72) × (4πR^2) = 2.88πR^2

Finally, we can express the volume of the egg using the following equation:

V = D = 2.88πR^2 × (depth of the submerged part in water)

Now, we know that V = D = 280,000 cubic centimeters and D is equal to 2.88πR^2 × (depth of the submerged part in water). By substituting the given values and solving this equation, we can find the total volume, V, of the egg.