Compute the order of magnitude of the mass of (a) a bathtub filled with water, and (b) a bathtub filled with pennies. In your solution list the quantities you estimate and the value you estimate for each.

The average "one person" bathtub holds about 80 gallons but is filled to only about 50 gallons. One gallon of water. weighs about 7.5 pounds.

If you filled it with pennies, the contents would weigh about 6 times as much. Remember that pennies are mostly zinc, with only a thin layer of copper.

They don't say if you are supposed to include the weight of the empty tub.

The weight of a typical empty bathtub, porcelain on steel, is about 350 lb.

To compute the order of magnitude of the mass of a bathtub filled with water and a bathtub filled with pennies, we will estimate the quantities involved and find the approximate values for each.

(a) Bathtub filled with water:
1. Estimate the dimensions of a typical bathtub: Let's say the length is 1.5 meters, the width is 0.75 meters, and the height is 0.45 meters.
2. Estimate the average depth when filled with water: Let's assume it's about 0.3 meters.
3. Calculate the volume of the bathtub: Multiply the length, width, and depth: V = 1.5 m * 0.75 m * 0.3 m = 0.3375 cubic meters.
4. Estimate the density of water: Water has a density of approximately 1000 kilograms per cubic meter.
5. Calculate the mass of water: Multiply the volume and density: M = V * density = 0.3375 m^3 * 1000 kg/m^3 = 337.5 kilograms.

The estimated mass of a bathtub filled with water is approximately 337.5 kilograms.

(b) Bathtub filled with pennies:
1. Estimate the dimensions of a typical bathtub: Let's use the same dimensions as in part (a).
2. Estimate the average thickness of a penny: Let's assume it's about 2 mm (0.002 meters).
3. Estimate the average density of a penny: Pennies are mainly made of copper, which has a density of approximately 8,960 kilograms per cubic meter.
4. Calculate the volume of a single penny: Multiply the length, width, and thickness: V_penny = 0.75 m * 0.75 m * 0.002 m = 0.001125 cubic meters.
5. Estimate the average number of pennies that could fit in the bathtub: To simplify the calculation, assume that pennies can be stacked without leaving any empty space. The volume of the bathtub can be divided by the volume of a single penny: Number_of_pennies = V_bathtub / V_penny = 0.3375 m^3 / 0.001125 m^3 = 300.
6. Calculate the mass of pennies: Multiply the number of pennies and the density of copper: M = Number_of_pennies * density = 300 * 8,960 kg/m^3 = 2,688,000 kilograms.

The estimated mass of a bathtub filled with pennies is approximately 2,688,000 kilograms.

In summary:
(a) The estimated mass of a bathtub filled with water is approximately 337.5 kilograms.
(b) The estimated mass of a bathtub filled with pennies is approximately 2,688,000 kilograms.