Answer: Let's use the formula for the density of a mixture to solve this problem:
ρ_mix = (m_A + m_B) / (V_A + V_B)
where ρ_mix is the density of the mixture, m_A and m_B are the masses of liquids A and B, and V_A and V_B are the volumes of liquids A and B.
Since we know the densities of liquids A and B, we can use them to calculate the masses of the liquids:
m_A = ρ_A * V_A
m_B = ρ_B * V_B
where ρ_A and ρ_B are the densities of liquids A and B.
We are given that the volume of liquid B is 50 cm³, so we can write:
V_B = 50 cm³
To solve for x, we need to find the value of V_A. Let's use the fact that x cm³ of liquid A are mixed with 50 cm³ of liquid B to write:
V_A + V_B = x + 50 cm³
Substituting the expressions for V_B and m_B into the formula for the density of the mixture, we get:
ρ_mix = (m_A + ρ_B * V_B) / (V_A + V_B)
Substituting the expressions for m_A and m_B, we get:
ρ_mix = (ρ_A * V_A + ρ_B * V_B) / (V_A + V_B)
We are given that the density of the resulting mixture is 2.7 g/cm³, so we can write:
ρ_mix = 2.7 g/cm³
Substituting all these values into the formula for the density of the mixture, we get:
2.7 g/cm³ = (ρ_A * V_A + 2.4 g/cm³ * 50 cm³) / (x + 50 cm³)
Simplifying this expression, we get:
2.7 g/cm³ = (ρ_A * V_A + 120 g) / (x + 50 cm³)
Multiplying both sides by (x + 50 cm³), we get:
2.7 g/cm³ * (x + 50 cm³) = ρ_A * V_A + 120 g
Expanding the left side, we get:
2.7 g/cm³ * x + 2.7 g/cm³ * 50 cm³ = ρ_A * V_A + 120 g
Simplifying this expression, we get:
ρ_A * V_A = 2.7 g/cm³ * x - 2.7 g/cm³ * 50 cm³ + 120 g
ρ_A * V_A = 2.7 g/cm³ * (x - 50 cm³) + 120 g
Now we can substitute the density of liquid A into this expression and solve for x:
ρ_A = 3.5 g/cm³
ρ_A * V_A = 2.7 g/cm³ * (x - 50 cm³) + 120 g
3.5 g/cm³ * V_A = 2.7 g/cm³ * (x - 50 cm³) + 120 g
3.5 g/cm³ * V_A = 2.7 g/cm³ * x - 2.7 g/cm³ * 50 cm³ + 120 g
3.5 g/cm³ * V_A = 2.7 g/cm³ * x - 54 g + 120 g
3.5 g/cm³ * V_A = 2.7 g/cm³ * x + 66 g
V_A = (2.7 g/cm³ *x + 66 g) / 3.5 g/cm³
V_A = (0.7714 x + 77.14) cm³/g
Now we can substitute this expression for V_A into the earlier equation we obtained for the density of the mixture, and solve for x:
2.7 g/cm³ = (ρ_A * V_A + 2.4 g/cm³ * 50 cm³) / (x + 50 cm³)
2.7 g/cm³ = (3.5 g/cm³ * (0.7714 x + 77.14) cm³/g + 2.4 g/cm³ * 50 cm³) / (x + 50 cm³)
2.7 g/cm³ = (2.6999 x + 227.6) / (x + 50 cm³)
Multiplying both sides by (x + 50 cm³), we get:
2.7 g/cm³ * (x + 50 cm³) = 2.6999 x + 227.6
Expanding the left side, we get:
2.7 g/cm³ * x + 2.7 g/cm³ * 50 cm³ = 2.6999 x + 227.6
Simplifying this expression, we get:
0.0001 x = 1.4
x = 14,000 cm³
Therefore, 14,000 cm³ of liquid A should be mixed with 50 cm³ of liquid B to obtain a mixture with density 2.7 g/cm³.