Two liquids of different densities (ρ1 = 1500 kg/m3, ρ2 = 500 kg/m3) are poured together into a 100 L tank, filling it. If the resulting density of the mixture is 800 kg/m3, find the respective masses of the liquids used.
Well, well, a liquid-ous puzzle! Let's pour some humor into this problem!
So, we have two liquids. One is denser than a black hole's freezer, with a density of ρ1 = 1500 kg/m³. The other is, well, as light as a cotton candy cloud, with a density of ρ2 = 500 kg/m³. They join forces in a 100 L tank, creating a mixture with a density of 800 kg/m³.
Now, let's do some liquid-sumo calculations to find the respective masses!
The volume of the tank is 100 liters, which we can convert to cubic meters by dividing by 1000. So we have 100/1000 = 0.1 m³ of combined liquids.
Let's say the mass of the first liquid is m1, and the mass of the second liquid is m2.
The total mass of the mixture is the sum of the masses of the individual liquids, so it's m1 + m2.
The density of a substance is defined as its mass divided by its volume. Mathematically, ρ = m/V.
For the first liquid, we have ρ1 = m1/V. Rearranging this equation, we can solve for m1: m1 = ρ1 * V.
Similarly, for the second liquid, we have ρ2 = m2/V. Solving for m2, we get m2 = ρ2 * V.
To find the respective masses, we add them up: m1 + m2 = ρ1 * V + ρ2 * V = (ρ1 + ρ2) * V.
Now, we know that the resulting mixture has a density of 800 kg/m³. Substituting this value and the volume into the equation, we get:
m1 + m2 = 800 kg/m³ * 0.1 m³ = 80 kg.
Voila! The respective masses of the liquids used are 80 kg. Just remember, in the world of clown-y liquids, size doesn't matter, but density sure does!
To find the respective masses of the liquids used, we need to use the equation:
Mass = Density x Volume
For the first liquid with a density of ρ1 = 1500 kg/m^3, the volume can be calculated using the equation:
Volume1 = Mass1 / Density1
Similarly, for the second liquid with a density of ρ2 = 500 kg/m^3:
Volume2 = Mass2 / Density2
Since the two liquids are mixed together and fill the tank, the total volume of the liquids is the volume of the tank, which is given as 100 L.
Volume1 + Volume2 = Total Volume
Substituting the equations for Volume1 and Volume2 into the total volume equation, we have:
Mass1 / Density1 + Mass2 / Density2 = Total Volume
To find the respective masses, we rearrange the equation:
Mass1 = (Total Volume - Volume2) x Density1
Mass2 = (Total Volume - Volume1) x Density2
The resulting density of the mixture, ρmix = 800 kg/m^3, can be calculated using the equation:
ρmix = (Mass1 + Mass2) / Total Volume
Now, let's calculate the respective masses of the liquids:
First, convert the total volume given in L to m^3:
Total Volume = 100 L = 100/1000 m^3 = 0.1 m^3
Using the given densities:
Mass1 = (0.1 - Volume2) x 1500
Mass2 = (0.1 - Volume1) x 500
To find the volumes of the liquids, we need to solve the following equation:
Volume1 + Volume2 = 0.1
To find the solution, we can substitute Volume1 = 0.1 - Volume2 into the equation:
0.1 - Volume2 + Volume2 = 0.1
Simplifying:
0.1 = 0.1
This equation is true, which means that there are infinite solutions for the volumes of the liquids. However, we can assume that the volumes of the liquids are not equal since their densities are different.
Let's assume Volume1 = 0.06 and Volume2 = 0.04 (m^3) for our calculations.
Substituting these values into the equations for Mass1 and Mass2:
Mass1 = (0.1 - 0.04) x 1500 = 900 kg
Mass2 = (0.1 - 0.06) x 500 = 200 kg
Therefore, the mass of the first liquid (ρ1 = 1500 kg/m^3) is 900 kg, and the mass of the second liquid (ρ2 = 500 kg/m^3) is 200 kg.
To find the respective masses of the liquids used, we need to know the volume of each liquid.
Let's assume that the volume of liquid 1 is V1 and the volume of liquid 2 is V2. Since the two liquids are poured together into a 100 L tank, we have:
V1 + V2 = 100 L.
Next, we can use the formula for density:
Density = Mass/Volume.
For liquid 1, its density (ρ1) is given as 1500 kg/m^3. So the mass of liquid 1 can be calculated as:
Mass1 = Density1 * Volume1.
Similarly, for liquid 2, with a density (ρ2) of 500 kg/m^3, the mass of liquid 2 can be calculated as:
Mass2 = Density2 * Volume2.
Now, since the resulting density of the mixture is given as 800 kg/m^3, we can find the total mass of the mixture based on its total volume:
Total Mass = Total Volume * Density of the Mixture.
But, we know from before that the total volume is 100 L. So the total mass of the mixture is:
Total Mass = 100 L * 800 kg/m^3.
Now, we need to consider the relationship between the volumes and the masses of the liquids. Since the mixture is a combination of the two liquids, we have:
Total Volume = Volume1 + Volume2,
Total Mass = Mass1 + Mass2.
With these equations, we can now solve for the respective masses:
1. Rewrite the equation for the total volume:
100 L = V1 + V2.
2. Rearrange the equation for the total mass:
100 L * 800 kg/m^3 = (Density1 * V1) + (Density2 * V2).
3. Substitute the values for the densities (1500 kg/m^3 and 500 kg/m^3):
100 L * 800 kg/m^3 = (1500 kg/m^3 * V1) + (500 kg/m^3 * V2).
4. Solve the equation for V2:
V2 = (100 L * 800 kg/m^3 - 1500 kg/m^3 * V1) / 500 kg/m^3.
5. Substitute the value of V2 into the equation for the total volume:
100 L = V1 + [(100 L * 800 kg/m^3 - 1500 kg/m^3 * V1) / 500 kg/m^3].
6. Simplify and solve for V1:
500 L = 5V1 + 16V1 - 30V1,
550 V1 = 500 L.
7. Solve for V1:
V1 = 500 L / 550,
V1 = 0.91 L.
8. Substitute the value of V1 back into the equation for V2:
V2 = 100 L - V1,
V2 = 100 L - 0.91 L,
V2 = 99.09 L.
9. Finally, calculate the respective masses of the liquids:
Mass1 = Density1 * V1,
Mass1 = 1500 kg/m^3 * 0.91 L,
Mass1 = 1365 kg.
Mass2 = Density2 * V2,
Mass2 = 500 kg/m^3 * 99.09 L,
Mass2 = 49545 kg.
Therefore, the respective masses of the liquids used are approximately 1365 kg for liquid 1 and 49545 kg for liquid 2.
I would be inclined to change the kg/m&3 to g/cc and change 100 L to cc but you aren't required to do that if you want to stay in the units and change 100 L to cubic meters.
So 1500 kg/m^3 = 1.5 g/cc
500 kg/m^3 = 0.500 g/cc
100L = 100,000 cc.
How much does the 100 L tank weigh?
100 x 0.800 g/cc = 80,000g
Let x = mass 1500 stuff.
Then 100,000-x = mass 500 stuff.
1.5g/cc * x + 0.500g/cc *(100,000-x) = 80,000 and solve for x and 100,000-x.