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Cylindrical container 1: Diameter D1 = 15 cm, Depth h1 = 20 cm

Mathematics

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Here's the solution for question 13 a) and c):

13 a)
Given:

  • Cylindrical container 1: Diameter D1=15D_1 = 15 cm, Depth h1=20h_1 = 20 cm
  • Cylindrical jar 2: Diameter D2=10D_2 = 10 cm
  1. Calculate the radius for each cylinder.

    For container 1: r1=D12=152=7.5r_1 = \frac{D_1}{2} = \frac{15}{2} = 7.5 cm
    For jar 2: r2=D22=102=5r_2 = \frac{D_2}{2} = \frac{10}{2} = 5 cm

  2. Calculate the volume of water in container 1.

    The volume of a cylinder is given by V=πr2hV = \pi r^2 h.

    V1=πr12h1V_1 = \pi r_1^2 h_1 V1=π(7.5 cm)2(20 cm)V_1 = \pi (7.5 \text{ cm})^2 (20 \text{ cm}) V1=π(56.25 cm2)(20 cm)V_1 = \pi (56.25 \text{ cm}^2) (20 \text{ cm}) V1=1125π cm3V_1 = 1125 \pi \text{ cm}^3
  3. Use the conservation of volume to find the depth in jar 2.

    When the water is poured into jar 2, its volume remains the same, so V2=V1V_2 = V_1.
    Let h2h_2 be the depth of water in jar 2.

    V2=πr22h2V_2 = \pi r_2^2 h_2 1125π cm3=π(5 cm)2h21125 \pi \text{ cm}^3 = \pi (5 \text{ cm})^2 h_2 1125π cm3=25π cm2h21125 \pi \text{ cm}^3 = 25 \pi \text{ cm}^2 h_2

    Divide both sides by 25π25 \pi:

    h2=1125π cm325π cm2h_2 = \frac{1125 \pi \text{ cm}^3}{25 \pi \text{ cm}^2} h2=112525 cmh_2 = \frac{1125}{25} \text{ cm} h2=45 cmh_2 = 45 \text{ cm}

    The depth of the water in the jar is 45 cm\boxed{\text{45 cm}}.

    13 c)
    Given:

    • Cylindrical tank: Diameter D=5.0D = 5.0 m, Volume of water V=110000V = 110 000 dm3^3
  4. Convert all measurements to consistent units.

    Let's convert everything to meters.
    Radius r=D2=5.0 m2=2.5r = \frac{D}{2} = \frac{5.0 \text{ m}}{2} = 2.5 m
    Volume V=110000V = 110 000 dm3^3. Since 1 m=10 dm1 \text{ m} = 10 \text{ dm}, then 1 m3=(10 dm)3=1000 dm31 \text{ m}^3 = (10 \text{ dm})^3 = 1000 \text{ dm}^3.

    V=110000 dm31000 dm3/m3=110 m3V = \frac{110 000 \text{ dm}^3}{1000 \text{ dm}^3/\text{m}^3} = 110 \text{ m}^3
  5. Use the volume formula to find the height of the water.

    The volume of a cylinder is V=πr2hV = \pi r^2 h. We need to find hh.
    Using π=227\pi = \frac{22}{7}:

    110 m3=227×(2.5 m)2×h110 \text{ m}^3 = \frac{22}{7} \times (2.5 \text{ m})^2 \times h 110 m3=227×6.25 m2×h110 \text{ m}^3 = \frac{22}{7} \times 6.25 \text{ m}^2 \times h h=110 m3×722×6.25 m2h = \frac{110 \text{ m}^3 \times 7}{22 \times 6.25 \text{ m}^2} h=770137.5 mh = \frac{770}{137.5} \text{ m} h=5.6 mh = 5.6 \text{ m}

    The height of the water in the tank is 5.6 m\boxed{\text{5.6 m}}.

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