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A cylindrical swimming pool has a diameter of 4.2m and a height of 1.7m. Calculate its volume in litres, given that 1000 cm³ = 1 L.

Mathematics

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Answers

3 parts
a

23554.7 L\text{23554.7 L}

b

36.29 m2\text{36.29 m}^2

c

R1995.25\text{R1995.25}

Step-by-step solution

10 steps

Here are the calculations for the swimming pool:

Part a· 4 steps

Calculate how many litres of water will fill the pool if 1000 cm3=1 L1000 \text{ cm}^3 = 1 \text{ L}.

  1. Identify the dimensions of the cylindrical pool.

    The diameter (DD) is 4.2 m4.2 \text{ m}.
    The radius (rr) is half of the diameter:

    r=D2=4.2 m2=2.1 mr = \frac{D}{2} = \frac{4.2 \text{ m}}{2} = 2.1 \text{ m}

    The height (hh) is 1.7 m1.7 \text{ m}.

  2. Calculate the volume of the cylinder in cubic meters.

    The formula for the volume of a cylinder is V=πr2hV = \pi r^2 h.

    V=π(2.1 m)2(1.7 m)V = \pi (2.1 \text{ m})^2 (1.7 \text{ m}) V=π(4.41 m2)(1.7 m)V = \pi (4.41 \text{ m}^2) (1.7 \text{ m}) V=7.497π m3V = 7.497\pi \text{ m}^3

    Using π≈3.14159\pi \approx 3.14159:

    V≈7.497×3.14159 m3V \approx 7.497 \times 3.14159 \text{ m}^3 V≈23.5547 m3V \approx 23.5547 \text{ m}^3
  3. Convert the volume from cubic meters to cubic centimeters.

    We know that 1 m=100 cm1 \text{ m} = 100 \text{ cm}, so 1 m3=(100 cm)3=1,000,000 cm31 \text{ m}^3 = (100 \text{ cm})^3 = 1,000,000 \text{ cm}^3.

    V in cm3=23.5547 m3×1,000,000cm3m3V \text{ in cm}^3 = 23.5547 \text{ m}^3 \times 1,000,000 \frac{\text{cm}^3}{\text{m}^3} V≈23,554,700 cm3V \approx 23,554,700 \text{ cm}^3
  4. Convert the volume from cubic centimeters to litres.

    Given that 1000 cm3=1 L1000 \text{ cm}^3 = 1 \text{ L}.

    V in L=23,554,700 cm31000cm3LV \text{ in L} = \frac{23,554,700 \text{ cm}^3}{1000 \frac{\text{cm}^3}{\text{L}}} V≈23,554.7 LV \approx 23,554.7 \text{ L}

    The volume of water that will fill the pool is 23554.7 L\boxed{\text{23554.7 L}}.

Part b

36.29 m2\text{36.29 m}^2

How many square metres of tiles are needed to tile the inside of the pool?

  1. Determine the areas to be tiled.

    The inside of the pool consists of the circular base and the lateral (curved) surface area.
    Area of the base (AbaseA_{base}) = πr2\pi r^2
    Lateral surface area (AlateralA_{lateral}) = 2πrh2\pi r h
    Total area (AtotalA_{total}) = Abase+AlateralA_{base} + A_{lateral}

  2. Calculate the area of the base.

    Abase=π(2.1 m)2A_{base} = \pi (2.1 \text{ m})^2 Abase=4.41π m2A_{base} = 4.41\pi \text{ m}^2 Abase≈4.41×3.14159 m2A_{base} \approx 4.41 \times 3.14159 \text{ m}^2 Abase≈13.854 m2A_{base} \approx 13.854 \text{ m}^2
  3. Calculate the lateral surface area.

    Alateral=2π(2.1 m)(1.7 m)A_{lateral} = 2\pi (2.1 \text{ m}) (1.7 \text{ m}) Alateral=7.14π m2A_{lateral} = 7.14\pi \text{ m}^2 Alateral≈7.14×3.14159 m2A_{lateral} \approx 7.14 \times 3.14159 \text{ m}^2 Alateral≈22.431 m2A_{lateral} \approx 22.431 \text{ m}^2
  4. Calculate the total area for tiling.

    Atotal=Abase+AlateralA_{total} = A_{base} + A_{lateral} Atotal=13.854 m2+22.431 m2A_{total} = 13.854 \text{ m}^2 + 22.431 \text{ m}^2 Atotal=36.285 m2A_{total} = 36.285 \text{ m}^2

    Rounding to two decimal places:
    The total area of tiles needed is 36.29 m2\boxed{\text{36.29 m}^2}.

Part c

R1995.25\text{R1995.25}

If the tiles cost R54,99/m2^2, calculate the cost of tiles for the pool.

  1. Use the total area calculated in part (b) and the given cost per square meter.

    Total area = 36.285 m236.285 \text{ m}^2
    Cost per square meter = R54.99/m254.99/\text{m}^2

  2. Calculate the total cost.

    Cost=Total area×Cost per square meter\text{Cost} = \text{Total area} \times \text{Cost per square meter} Cost=36.285 m2×R54.99/m2\text{Cost} = 36.285 \text{ m}^2 \times \text{R}54.99/\text{m}^2 Cost≈R1995.25215\text{Cost} \approx \text{R}1995.25215

    Rounding to two decimal places for currency:
    The cost of tiles for the pool is R1995.25\boxed{\text{R1995.25}}.

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