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A toy is in the form of a hemisphere surmounted by a cone. The height of the cone is 2 cm and the diameter of its base is 4 cm. Find the volume of the toy. Use pi = 22/7.

Mathematics

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Here is the solution for question 36.

  1. Identify the given dimensions and formulas.

    The toy consists of a hemisphere surmounted by a cone.
    Height of the cone (hh) = 2 cm.
    Diameter of the base = 4 cm.
    Radius (rr) = diameter2=4 cm2=2\frac{\text{diameter}}{2} = \frac{4 \text{ cm}}{2} = 2 cm.
    We are given to use π=227\pi = \frac{22}{7}.

  2. Calculate the volume of the toy.

    The total volume of the toy is the sum of the volume of the hemisphere and the volume of the cone.
    Volume of hemisphere (VhemisphereV_{hemisphere}) = 23πr3\frac{2}{3} \pi r^3
    Volume of cone (VconeV_{cone}) = 13πr2h\frac{1}{3} \pi r^2 h

    Calculate VhemisphereV_{hemisphere}:

    Vhemisphere=23×227×(2 cm)3V_{hemisphere} = \frac{2}{3} \times \frac{22}{7} \times (2 \text{ cm})^3 Vhemisphere=23×227×8 cm3V_{hemisphere} = \frac{2}{3} \times \frac{22}{7} \times 8 \text{ cm}^3 Vhemisphere=35221 cm3V_{hemisphere} = \frac{352}{21} \text{ cm}^3

    Calculate VconeV_{cone}:

    Vcone=13×227×(2 cm)2×2 cmV_{cone} = \frac{1}{3} \times \frac{22}{7} \times (2 \text{ cm})^2 \times 2 \text{ cm} Vcone=13×227×4 cm2×2 cmV_{cone} = \frac{1}{3} \times \frac{22}{7} \times 4 \text{ cm}^2 \times 2 \text{ cm} Vcone=17621 cm3V_{cone} = \frac{176}{21} \text{ cm}^3

    Calculate the total volume (VtoyV_{toy}):

    Vtoy=Vhemisphere+VconeV_{toy} = V_{hemisphere} + V_{cone} Vtoy=35221 cm3+17621 cm3V_{toy} = \frac{352}{21} \text{ cm}^3 + \frac{176}{21} \text{ cm}^3 Vtoy=352+17621 cm3V_{toy} = \frac{352 + 176}{21} \text{ cm}^3 Vtoy=52821 cm3V_{toy} = \frac{528}{21} \text{ cm}^3

    Simplify the fraction by dividing numerator and denominator by 3:

    Vtoy=1767 cm3V_{toy} = \frac{176}{7} \text{ cm}^3

    The volume of the toy is 1767 cm3\boxed{\frac{176}{7} \text{ cm}^3}.

  3. Calculate the surface area of the toy.

    The total surface area of the toy is the sum of the curved surface area of the hemisphere and the curved surface area of the cone.
    Curved surface area of hemisphere (CSAhemisphereCSA_{hemisphere}) = 2πr22 \pi r^2
    Curved surface area of cone (CSAconeCSA_{cone}) = πrl\pi r l, where ll is the slant height of the cone.

    First, calculate the slant height (ll) of the cone:

    l=r2+h2l = \sqrt{r^2 + h^2} l=(2 cm)2+(2 cm)2l = \sqrt{(2 \text{ cm})^2 + (2 \text{ cm})^2} l=4 cm2+4 cm2l = \sqrt{4 \text{ cm}^2 + 4 \text{ cm}^2} l=8 cm2l = \sqrt{8 \text{ cm}^2} l=22 cml = 2\sqrt{2} \text{ cm}

    Calculate CSAhemisphereCSA_{hemisphere}:

    CSAhemisphere=2×227×(2 cm)2CSA_{hemisphere} = 2 \times \frac{22}{7} \times (2 \text{ cm})^2 CSAhemisphere=2×227×4 cm2CSA_{hemisphere} = 2 \times \frac{22}{7} \times 4 \text{ cm}^2 CSAhemisphere=1767 cm2CSA_{hemisphere} = \frac{176}{7} \text{ cm}^2

    Calculate CSAconeCSA_{cone}:

    CSAcone=227×2 cm×22 cmCSA_{cone} = \frac{22}{7} \times 2 \text{ cm} \times 2\sqrt{2} \text{ cm} CSAcone=8827 cm2CSA_{cone} = \frac{88\sqrt{2}}{7} \text{ cm}^2

    Calculate the total surface area (SAtoySA_{toy}):

    SAtoy=CSAhemisphere+CSAconeSA_{toy} = CSA_{hemisphere} + CSA_{cone} SAtoy=1767 cm2+8827 cm2SA_{toy} = \frac{176}{7} \text{ cm}^2 + \frac{88\sqrt{2}}{7} \text{ cm}^2 SAtoy=176+8827 cm2SA_{toy} = \frac{176 + 88\sqrt{2}}{7} \text{ cm}^2

    Factor out 88 from the numerator:

    SAtoy=88(2+2)7 cm2SA_{toy} = \frac{88(2 + \sqrt{2})}{7} \text{ cm}^2

    The surface area of the toy is 88(2+2)7 cm2\boxed{\frac{88(2 + \sqrt{2})}{7} \text{ cm}^2}.

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