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9. A metal hemispherical bowl has an external diameter of 50.8 cm and a thickness of 2.54 cm.

(i) Given that the empty bowl weighs 97.9 kg, find the density, in kg/m³, of the metal from which the bowl is made.

(ii) If the bowl is completely filled with a liquid of density 31.75 kg/m³, find the mass of the liquid in grams.

Answer :

To solve this problem, we need to calculate two things:

Part (i)

Finding the Density of the Metal:

  1. First, let's determine the volume of the metal comprising the bowl.

  2. The external diameter of the bowl is given as 50.8 cm. Thus, the external radius is:
    [tex]\text{External Radius} = \frac{50.8}{2} = 25.4 \text{ cm}[/tex]

  3. The thickness of the bowl is 2.54 cm, so the internal radius is:
    [tex]\text{Internal Radius} = 25.4 - 2.54 = 22.86 \text{ cm}[/tex]

  4. Convert these measurements from centimeters to meters because density is given in kg/m³. So:

    • External radius = 0.254 m
    • Internal radius = 0.2286 m

  5. Calculate the volume of the external hemisphere:
    [tex]V_{\text{external}} = \frac{2}{3} \pi (0.254)^3[/tex]

  6. Calculate the volume of the internal hemisphere:
    [tex]V_{\text{internal}} = \frac{2}{3} \pi (0.2286)^3[/tex]

  7. The volume of the metal of the bowl is the difference between the external and internal hemispheres:
    [tex]V_{\text{metal}} = V_{\text{external}} - V_{\text{internal}}[/tex]

  8. Using these calculations:
    [tex]V_{\text{external}} = \frac{2}{3} \pi (0.254)^3 \approx 0.034358 \text{ m}^3[/tex]
    [tex]V_{\text{internal}} = \frac{2}{3} \pi (0.2286)^3 \approx 0.025032 \text{ m}^3[/tex]
    [tex]V_{\text{metal}} = 0.034358 - 0.025032 = 0.009326 \text{ m}^3[/tex]

  9. Given the weight of the bowl is 97.9 kg, use the formula for density:
    [tex]\text{Density} = \frac{\text{Mass}}{\text{Volume}}[/tex]
    [tex]\text{Density of the metal} = \frac{97.9}{0.009326} \approx 10496.3 \text{ kg/m}^3[/tex]

Part (ii)

Finding the Mass of the Liquid:

  1. Compute the volume of the liquid that fills the internal hemisphere of the bowl. The internal radius has already been calculated as 0.2286 m.

  2. Calculate the volume of the internal hemisphere:
    [tex]V_{\text{liquid}} = \frac{2}{3} \pi (0.2286)^3 \approx 0.025032 \text{ m}^3[/tex]

  3. Given the density of the liquid is 31.75 kg/m³, find the mass of the liquid:
    [tex]\text{Mass of the liquid} = \text{Density} \times \text{Volume}[/tex]
    [tex]\text{Mass of the liquid} = 31.75 \times 0.025032 \approx 0.795 \text{ kg}[/tex]

  4. Convert the mass of the liquid to grams (since 1 kg = 1000 grams):
    [tex]\text{Mass of the liquid in grams} = 0.795 \times 1000 = 795 \text{ grams}[/tex]

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