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A cylindrical tank has a radius of 1.4 m and a height of 2.4 m. (d) What is the volume of water needed to fill the tank in litres? (e) How many buckets are needed to fill the tank?

Mathematics

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2 parts
d

14778.1 litres\text{14778.1 litres}

e

6000\text{6000}

Step-by-step solution

6 steps

Here are the calculations for the volume of water and the number of buckets.

Part d· 4 steps

The volume of water required to fill the tank in litres.

  1. Convert the tank dimensions from meters to centimeters.

    Tank radius (RTR_T) = 1.4 m=1.4×100 cm=140 cm1.4 \text{ m} = 1.4 \times 100 \text{ cm} = 140 \text{ cm}
    Tank height (HTH_T) = 2.4 m=2.4×100 cm=240 cm2.4 \text{ m} = 2.4 \times 100 \text{ cm} = 240 \text{ cm}

  2. Calculate the volume of the cylindrical tank in cubic centimeters.

    The formula for the volume of a cylinder is V=πR2HV = \pi R^2 H.

    VT=π(140 cm)2(240 cm)V_T = \pi (140 \text{ cm})^2 (240 \text{ cm}) VT=π(19600 cm2)(240 cm)V_T = \pi (19600 \text{ cm}^2) (240 \text{ cm}) VT=4704000π cm3V_T = 4704000 \pi \text{ cm}^3

  3. Convert the volume from cubic centimeters to litres.

    Since 1 litre=1000 cm31 \text{ litre} = 1000 \text{ cm}^3:

    VT(litres)=4704000π cm31000 cm3/litreV_T (\text{litres}) = \frac{4704000 \pi \text{ cm}^3}{1000 \text{ cm}^3/\text{litre}} VT(litres)=4704π litresV_T (\text{litres}) = 4704 \pi \text{ litres}

    Using π≈3.14159\pi \approx 3.14159:

    VT(litres)≈4704×3.14159 litresV_T (\text{litres}) \approx 4704 \times 3.14159 \text{ litres} VT(litres)≈14778.1056 litresV_T (\text{litres}) \approx 14778.1056 \text{ litres}

  4. Round the answer to one decimal place.

    VT≈14778.1 litresV_T \approx 14778.1 \text{ litres}

    The volume of water required to fill the tank is 14778.1 litres\boxed{\text{14778.1 litres}}.

Part e

6000\text{6000}

Calculate the number of buckets that must be drawn to fill the tank.

  1. Calculate the volume of one bucket (frustum) in cubic centimeters.

    The formula for the volume of a frustum is Vf=13πhf(R2+Rr+r2)V_f = \frac{1}{3} \pi h_f (R^2 + Rr + r^2).
    Bucket depth (hfh_f) = 12 cm
    Top radius (RR) = 10 cm
    Bottom radius (rr) = 6 cm

    VB=13π(12 cm)((10 cm)2+(10 cm)(6 cm)+(6 cm)2)V_B = \frac{1}{3} \pi (12 \text{ cm}) ((10 \text{ cm})^2 + (10 \text{ cm})(6 \text{ cm}) + (6 \text{ cm})^2) VB=4π cm(100 cm2+60 cm2+36 cm2)V_B = 4 \pi \text{ cm} (100 \text{ cm}^2 + 60 \text{ cm}^2 + 36 \text{ cm}^2) VB=4π cm(196 cm2)V_B = 4 \pi \text{ cm} (196 \text{ cm}^2) VB=784π cm3V_B = 784 \pi \text{ cm}^3

  2. Calculate the number of buckets required by dividing the total tank volume by the volume of one bucket.

    N=Volume of tankVolume of one bucketN = \frac{\text{Volume of tank}}{\text{Volume of one bucket}} N=4704000π cm3784π cm3N = \frac{4704000 \pi \text{ cm}^3}{784 \pi \text{ cm}^3}

    The π\pi terms cancel out:

    N=4704000784N = \frac{4704000}{784} N=6000N = 6000

    The number of buckets that must be drawn to fill the tank is 6000\boxed{\text{6000}}.

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