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Surface Areas and Volumes: Competency Practice

Pharmaceutical capsule capacity, solid immersion water displacement, and composite solid surface areas.

6 problems·20–25 min·★★★★☆
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  1. Problem 1
    surface-areas-and-volumes·★★★★☆
    A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The length of the entire capsule is $12\text{ mm}$ and the diameter of the capsule is $4\text{ mm}$. (i) Find the length of the cylindrical portion of the capsule. (ii) Find the total surface area of the capsule.
    ▶Answer
    (i) $8\text{ mm}$, (ii) $48\pi\text{ mm}^2$
    ▶Step-by-step solution
    1. (i) Radius of hemispheres and cylinder $r = \frac{4}{2} = 2\text{ mm}$. Length of cylindrical portion $h = 12 - 2r = 8\text{ mm}$.
    2. (ii) Total Surface Area $= \text{CSA of cylinder} + 2 \times \text{CSA of hemisphere} = 2\pi rh + 4\pi r^2 = 2\pi r(h + 2r) = 2\pi r L = 2\pi (2)(12) = 48\pi\text{ mm}^2$.
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  2. Problem 2
    volume·★★★★★
    A cylinder of radius 6 cm and height 15 cm is full of water. A solid cone of height 8 cm joined to a hemisphere, both radius 3 cm, is immersed completely. Find the water remaining.
    ▶Answer
    $498\pi$ cm$^3$
    ▶Step-by-step solution
    1. Cylinder volume is $540\pi$ cm$^3$.
    2. The immersed solid displaces $42\pi$ cm$^3$; subtract.
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  3. Problem 3
    surface-area·★★★★☆
    A cone of radius 9 cm and height 12 cm is joined to a hemisphere of the same radius. Find the exposed surface area.
    ▶Answer
    $297\pi$ cm$^2$
    ▶Step-by-step solution
    1. The cone's slant height is $\sqrt{9^2+12^2}=15$ cm.
    2. Add $\pi rl+2\pi r^2=297\pi$ cm$^2$.
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  4. Problem 4
    surface-area·★★★★☆
    A hollow cylindrical cup of radius 7 cm and height 10 cm has a hollow hemispherical bowl of the same radius fixed at one end. Find its inner surface area.
    ▶Answer
    $238\pi$ cm$^2$
    ▶Step-by-step solution
    1. Add only the inner curved surfaces.
    2. $2\pi rh+2\pi r^2=238\pi$ cm$^2$.
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  5. Problem 5
    surface-area·★★★★☆
    A hemisphere of the greatest possible radius is mounted on a cube of edge 8 cm. Find the exposed surface area.
    ▶Answer
    $384+16\pi$ cm$^2$
    ▶Step-by-step solution
    1. The greatest radius is 4 cm.
    2. Subtract the covered base circle from the cube and add the hemisphere's curved area, leaving a net $+\pi r^2$.
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  6. Problem 6
    surface-area·★★★★☆
    A hemisphere is scooped from each flat end of a cylinder of radius 3 cm and height 20 cm. Find the exposed surface area.
    ▶Answer
    $156\pi$ cm$^2$
    ▶Step-by-step solution
    1. The exposed parts are the cylinder's curved surface and two hemispherical cavities.
    2. $2\pi rh+4\pi r^2=156\pi$ cm$^2$.
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