# Density Problems

Problem # 1

A solid ball has a mass of 50 grams and a volume of 20 cm3. What is the density? (Answer: 2.5 g/cm3)

Problem # 2

A solid ball has a mass of 100 grams and a radius of 2 cm. What is the density? (Answer: 2.98 g/cm3)

Problem # 3

A solid cylinder has a radius of 2 cm and a length of 7 cm. It has a density of 3.1 g/cm3. What is the mass of the cylinder? (Answer: 272.69 grams)

Problem # 4

Seawater contains approximately 3.5% salt by weight. How much seawater (in kg) contains 1 kg of salt? If the density of seawater is 1030 kg/m3, what is the volume of seawater, in liters, containing 1 kg of salt? (Answer: 28.57 kg, 27.74 L)

Problem # 5

A solid cylinder of length 10 cm is placed in water. It stands upright with the top 3 cm protruding above the water surface. The density of water is 1.0 g/cm3. What is the density of the cylinder?

Problem # 6

The density of two liquids (A and B) is given as 1000 kg/m3 and 600 kg/m3, respectively. The two liquids are mixed in a certain proportion and the density of the resulting liquid is 850 kg/m3. How much of liquid B (in grams) does 1 kg of the mixture contain? Assume the volume of the two liquids is additive when mixed.

Problem # 7

It is given that 10 mL of fuel stabilizer treats 3 L of gasoline. Fuel stabilizer is added, in the specified concentration, to 30 liters of gasoline inside a tank to preserve it during storage. After some time has passed, 10 liters of gas is used up, and a fresh 10 liters of gas is pumped into the tank to replace it, but no additional stabilizer is added. After some additional time has passed, 15 liters of gas is used up and a fresh 15 liters of gas is pumped into the tank to replace it. How much stabilizer must now be added to the fuel inside the tank to maintain the correct concentration of stabilizer? Where appropriate, ignore the small volume contribution of the stabilizer in the calculations.

Hints And Numerical Answers For Density Problems

Hint and answer for Problem # 5

Use Archimedes' principle. Equate the mass of the cylinder to the mass of the water displaced by the cylinder. Mass = volume × density. For the cylinder, volume = (cross-sectional area) × length. For the water, volume = (cross-sectional area)×7. The cross-sectional area cancels out and we can easily calculate the density of the cylinder. The density is 0.7 g/cm3.

Hint and answer for Problem # 6

First, let VA represent the volume of liquid A, and let VB represent the volume of liquid B, in 1 kg of mixture. Then, 850×(VA+VB) = 1 kg (of mixture). In addition, 1000VA+600VB = 1 kg (of mixture). We have here two equations with two unknowns (VA and VB). Solve for VB. The mass of liquid B in 1 kg of mixture is 600VB, which is equal to 264.7 grams.

Hint and answer for Problem # 7

This is a nice variation on the usual density problems. It involves looking at the problem from the point of view of concentration. The initial amount of stabilizer in the tank is 100 mL. After 10 liters of gas is used up the first time, we have 20 liters remaining inside the tank. This corresponds to 2/3 of the fuel remaining, which also corresponds to 2/3 of the stabilizer remaining, which is 100×2/3 = 66.67 mL. Next, 10 liters of fresh gas is added, but no additional stabilizer is added, hence the amount of stabilizer remains at 66.67 mL (with 30 liters of gas now inside the tank). Next, 15 liters of gas is used up, which translates into 15 liters left in the tank, or 1/2 of the fuel remaining. This corresponds to 1/2 of the stabilizer remaining, which is 66.67×1/2 = 33.33 mL. We now add 15 liters of gas to the tank, bringing it back up to 30 liters, but with only 33.33 mL of stabilizer left. We know that for 30 liters of fuel we must add 100 mL of stabilizer. Hence, the amount of stabilizer to add is 100−33.33 = 66.67 mL.
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