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A bucket in the shape of a frustum of a cone has a top diameter of 28 cm, a bottom diameter of 21 cm, and a height of 20 cm. Calculate its capacity in litres, rounded to one decimal place.

Mathematics

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Answer

9.5 litres\text{9.5 litres}

Step-by-step solution

4 steps

  1. Identify the given dimensions and calculate the radii.

    The top diameter D=28D = 28 cm, so the top radius R=282=14R = \frac{28}{2} = 14 cm.
    The bottom diameter d=21d = 21 cm, so the bottom radius r=212=10.5r = \frac{21}{2} = 10.5 cm.
    The height of the frustum H=20H = 20 cm.

  2. Apply the formula for the volume of a frustum.

    The volume VV of a frustum is given by the formula:

    V=13πH(R2+Rr+r2)V = \frac{1}{3} \pi H (R^2 + Rr + r^2)

    Substitute the values:

    V=13π(20)(142+(14)(10.5)+10.52)V = \frac{1}{3} \pi (20) (14^2 + (14)(10.5) + 10.5^2)

  3. Calculate the terms inside the parenthesis and the volume in cm3^3.

    142=19614^2 = 196 (14)(10.5)=147(14)(10.5) = 147 10.52=110.2510.5^2 = 110.25

    Now, sum these values:

    R2+Rr+r2=196+147+110.25=453.25R^2 + Rr + r^2 = 196 + 147 + 110.25 = 453.25

    Substitute this back into the volume formula:

    V=13π(20)(453.25)V = \frac{1}{3} \pi (20) (453.25) V=90653πV = \frac{9065}{3} \pi

    Using π≈3.14159\pi \approx 3.14159:

    V≈90653×3.14159V \approx \frac{9065}{3} \times 3.14159 V≈3021.6666×3.14159V \approx 3021.6666 \times 3.14159 V≈9499.791 cm3V \approx 9499.791 \text{ cm}^3

  4. Convert the volume from cm3^3 to litres and round to one decimal place.

    Since 1 litre=1000 cm31 \text{ litre} = 1000 \text{ cm}^3, divide the volume in cm3^3 by 1000:

    Vlitres=9499.7911000 litresV_{\text{litres}} = \frac{9499.791}{1000} \text{ litres} Vlitres≈9.499791 litresV_{\text{litres}} \approx 9.499791 \text{ litres}

    Rounding to one decimal place:

    Vlitres≈9.5 litresV_{\text{litres}} \approx 9.5 \text{ litres}

    The capacity of the bucket is 9.5 litres\text{9.5 litres}.

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