Displacement Dilemma

BONUS QUESTION: How would your answer change if the container held a liquid that had a density of 1.5 grams per milliliter or .75 grams per milliliter?

Simple Explanation It can be observed that the sphere will fit inside the cylinder and the cylinder will fit inside the cube, both with space not occupied, therefore the cube has the most total volume and the sphere has the least.

The key is what happens when they are placed in the water as the cube floats, the cylinder has neutral buoyancy, and the sphere sinks. Because of this, the sphere displaces its entire volume (65.450 ml), the cylinder displaces its entire volume (98.175 ml) which equals its weight (98.175 g), and the cube displaces only that portion of its volume that is under water due to it's weight (98.175 ml), therefore the cube and cylinder displace the same amount and more than the sphere.

Detailed Explaination First calculate the volume of each shape as follows:

• Cube V = W x D x H = 50 x 50 x 50 = 125,000 mm $^{3}$ or 125 ml
• Cylinder V = $\pi R^{2} H$ = $\pi 25^{2}$ x 50 = 98174.770 mm $^{3}$ or 98.175 ml
• Sphere V = $\frac{4}{3} \pi R^{3}$ = $\frac{4}{3} \pi 25^{3}$ = 65449.847 mm $^{3}$ or 65.450 ml

Next, determine if the shape will sink or float by subtracting the weight of the water the shape would displace (if submerged) from the weight (mass) of the shapes.

• The cube displaces 125 ml of water which weighs 125 grams. Subtract cube mass of 98.175 grams leaving 26.825 grams (125 - 98.175 = 26.825). Since this is a positive number the shape would float and displace only 98.175 ml of water.
• The cylinder displaces 98.175 ml of water which weighs 98.175 grams. Subtract the cylinder mass of 98.175 grams leaving 0 grams indicating that the shape will have neutral buoyancy and will displace 98.175 ml of water, the same as the cube.
• The sphere displaces 65.450 ml of water which weighs 65.450 g. Subtract the sphere mass of 98.175 g leaving -32.725 grams (65.450-98.175 = -32.725). Since this is a negative number the shape will float and displace its entire volume of 65.450 ml of water which is less than the cube and the cylinder.

Therefore: the cube and the cylinder each displace 98.175 ml of water, more than the 65.450 ml that the sphere displaces.