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Equate the volume of the hemisphere to the volume of the water displaced in the cylinder.

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15. A cylinder has a radius of 10.5cm. A hemispherical solid is fully immersed in the cylinder containing water and the level water rose by a height of 6.2cm. Calculate the radius of the hemisphere to 1 d.p.

  1. Equate the volume of the hemisphere to the volume of the water displaced in the cylinder.

    The volume of water displaced is equal to the volume of the hemisphere.
    Volume of hemisphere (VhV_h) = 23πrh3\frac{2}{3}\pi r_h^3, where rhr_h is the radius of the hemisphere.
    Volume of water displaced in cylinder (VcV_c) = πRc2hw\pi R_c^2 h_w, where RcR_c is the radius of the cylinder and hwh_w is the height the water level rose.

    Vh=VcV_h = V_c 23πrh3=πRc2hw\frac{2}{3}\pi r_h^3 = \pi R_c^2 h_w
  2. Substitute the given values into the equation.

    Given Rc=10.5R_c = 10.5 cm and hw=6.2h_w = 6.2 cm.

    23πrh3=π(10.5)2(6.2)\frac{2}{3}\pi r_h^3 = \pi (10.5)^2 (6.2)
  3. Solve for rhr_h.

    Cancel π\pi from both sides:

    23rh3=(10.5)2(6.2)\frac{2}{3} r_h^3 = (10.5)^2 (6.2) 23rh3=110.25×6.2\frac{2}{3} r_h^3 = 110.25 \times 6.2 23rh3=683.55\frac{2}{3} r_h^3 = 683.55

    Multiply both sides by 32\frac{3}{2}:

    rh3=683.55×32r_h^3 = \frac{683.55 \times 3}{2} rh3=2050.652r_h^3 = \frac{2050.65}{2} rh3=1025.325r_h^3 = 1025.325

    Take the cube root of both sides:

    rh=1025.3253r_h = \sqrt[3]{1025.325} rh≈10.0838 cmr_h \approx 10.0838 \text{ cm}
  4. Round the radius to 1 decimal place.

    rh≈10.1 cmr_h \approx 10.1 \text{ cm}

    The radius of the hemisphere is 10.1 cm\boxed{10.1 \text{ cm}}.

    16. Complete the figure below to show rotational order 3.

    To complete the figure to show rotational order 3, the final shape must look identical after a rotation of 360∘/3=120∘360^\circ / 3 = 120^\circ about its center. The small dot in the figure indicates the center of rotation.

    The given figure shows one "point" or "arm" of a three-pointed star. To complete it, you need to draw two more identical points, each rotated by 120∘120^\circ from the previous one around the central dot.

    The completed figure will be a three-pointed star, where each point is identical to the one shown and is positioned at 120∘120^\circ intervals around the center.

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