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A mixture contains 200 cm³ of water and 300 cm³ of milk. The density of water is 1 g/cm³ and the density of milk is 2 g/cm³. Calculate (a) the total volume of the mixture, (b) the total mass of the mi

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3 parts
i

500 cm3500 \text{ cm}^3

ii

800 g800 \text{ g}

iii

1600 kg/m31600 \text{ kg/m}^3

481.25 cm3481.25 \text{ cm}^3

Step-by-step solution

11 steps

24. (a)

Part i· 2 steps

The volume of the mixture.

  1. Identify the given volumes.

    Volume of water (VwV_w) = 200 cm3200 \text{ cm}^3
    Volume of milk (VmV_m) = 300 cm3300 \text{ cm}^3

  2. Calculate the total volume of the mixture.

    The volumes are additive.

    Vmixture=Vw+VmV_{mixture} = V_w + V_m Vmixture=200 cm3+300 cm3V_{mixture} = 200 \text{ cm}^3 + 300 \text{ cm}^3 Vmixture=500 cm3V_{mixture} = \boxed{500 \text{ cm}^3}

Part ii

800 g800 \text{ g}

The total mass of the mixture.

  1. Convert densities to a consistent unit (e.g., g/cm3^3) or volumes to m3^3.

    Let's use g/cm3^3.
    Density of water (ρw\rho_w) = 1000 kg/m3=1 g/cm31000 \text{ kg/m}^3 = 1 \text{ g/cm}^3
    Density of milk (ρm\rho_m) = 2000 kg/m3=2 g/cm32000 \text{ kg/m}^3 = 2 \text{ g/cm}^3

  2. Calculate the mass of water.

    Mw=ρw×VwM_w = \rho_w \times V_w Mw=1 g/cm3×200 cm3M_w = 1 \text{ g/cm}^3 \times 200 \text{ cm}^3 Mw=200 gM_w = 200 \text{ g}

  3. Calculate the mass of milk.

    Mm=ρm×VmM_m = \rho_m \times V_m Mm=2 g/cm3×300 cm3M_m = 2 \text{ g/cm}^3 \times 300 \text{ cm}^3 Mm=600 gM_m = 600 \text{ g}

  4. Calculate the total mass of the mixture.

    Mmixture=Mw+MmM_{mixture} = M_w + M_m Mmixture=200 g+600 gM_{mixture} = 200 \text{ g} + 600 \text{ g} Mmixture=800 gM_{mixture} = \boxed{800 \text{ g}}

Part iii

1600 kg/m31600 \text{ kg/m}^3

The density of the mixture.

  1. Use the total mass and total volume of the mixture.

    Total mass (MmixtureM_{mixture}) = 800 g800 \text{ g}
    Total volume (VmixtureV_{mixture}) = 500 cm3500 \text{ cm}^3

  2. Calculate the density of the mixture.

    ρmixture=MmixtureVmixture\rho_{mixture} = \frac{M_{mixture}}{V_{mixture}} ρmixture=800 g500 cm3\rho_{mixture} = \frac{800 \text{ g}}{500 \text{ cm}^3} ρmixture=1.6 g/cm3\rho_{mixture} = 1.6 \text{ g/cm}^3

  3. Convert the density to kg/m3^3 if desired (as original densities were in kg/m3^3).

    1.6 g/cm3=1.6×1 kg1000 g×(100 cm)3(1 m)31.6 \text{ g/cm}^3 = 1.6 \times \frac{1 \text{ kg}}{1000 \text{ g}} \times \frac{(100 \text{ cm})^3}{(1 \text{ m})^3} 1.6 g/cm3=1.6×11000×1,000,000 kg/m31.6 \text{ g/cm}^3 = 1.6 \times \frac{1}{1000} \times 1,000,000 \text{ kg/m}^3 ρmixture=1600 kg/m3\rho_{mixture} = \boxed{1600 \text{ kg/m}^3}

    24. (b) Calculate the volume of a cylindrical measuring cylinder of radius 3.5 cm and height 12.5 cm.

  4. Identify the given dimensions.

    Radius (rr) = 3.5 cm3.5 \text{ cm}
    Height (hh) = 12.5 cm12.5 \text{ cm}

  5. Use the formula for the volume of a cylinder (V=πr2hV = \pi r^2 h).

    Using π=227\pi = \frac{22}{7}:

    V=πr2hV = \pi r^2 h V=227×(3.5 cm)2×12.5 cmV = \frac{22}{7} \times (3.5 \text{ cm})^2 \times 12.5 \text{ cm} V=227×12.25 cm2×12.5 cmV = \frac{22}{7} \times 12.25 \text{ cm}^2 \times 12.5 \text{ cm} V=22×1.75 cm2×12.5 cmV = 22 \times 1.75 \text{ cm}^2 \times 12.5 \text{ cm} V=38.5 cm2×12.5 cmV = 38.5 \text{ cm}^2 \times 12.5 \text{ cm} V=481.25 cm3V = \boxed{481.25 \text{ cm}^3}
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