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A salt shaker is made up of a cylinder of height h and a hemisphere of internal diameter d. Write down an expression for the volume of the salt shaker in terms of π, d and h. Find the internal volume of the salt shaker if h = 11 cm and d = 3,5 cm, leaving the answer in terms of π.

Mathematics

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Answers

2 parts
a

V=112πd2(3h+d)V = \frac{1}{12} \pi d^2 (3h + d)

b

V=357796π cm3V = \frac{3577}{96} \pi \text{ cm}^3

Step-by-step solution

6 steps

Part a· 4 steps

  1. Identify the radius of the cylinder and hemisphere.

    The diameter is dd, so the radius rr is d2\frac{d}{2}.

  2. Write the formula for the volume of the cylinder.

    The volume of a cylinder is Vcylinder=πr2hV_{\text{cylinder}} = \pi r^2 h. Substituting r=d2r = \frac{d}{2}:

    Vcylinder=π(d2)2h=πd24hV_{\text{cylinder}} = \pi \left(\frac{d}{2}\right)^2 h = \pi \frac{d^2}{4} h

  3. Write the formula for the volume of the hemisphere.

    The volume of a sphere is given as Vsphere=43πr3V_{\text{sphere}} = \frac{4}{3} \pi r^3. The volume of a hemisphere is half of this:

    Vhemisphere=12(43πr3)=23πr3V_{\text{hemisphere}} = \frac{1}{2} \left(\frac{4}{3} \pi r^3\right) = \frac{2}{3} \pi r^3

    Substituting r=d2r = \frac{d}{2}:

    Vhemisphere=23π(d2)3=23πd38=112πd3V_{\text{hemisphere}} = \frac{2}{3} \pi \left(\frac{d}{2}\right)^3 = \frac{2}{3} \pi \frac{d^3}{8} = \frac{1}{12} \pi d^3

  4. Sum the volumes to find the total volume of the salt shaker.

    Vtotal=Vcylinder+VhemisphereV_{\text{total}} = V_{\text{cylinder}} + V_{\text{hemisphere}} Vtotal=14πd2h+112πd3V_{\text{total}} = \frac{1}{4} \pi d^2 h + \frac{1}{12} \pi d^3

    Factor out common terms:

    Vtotal=112πd2(3h+d)V_{\text{total}} = \frac{1}{12} \pi d^2 (3h + d)

    The expression for the volume is:

    V=112πd2(3h+d)\boxed{V = \frac{1}{12} \pi d^2 (3h + d)}

Part b

V=357796π cm3V = \frac{3577}{96} \pi \text{ cm}^3

  1. Substitute the given values h=11 cmh = 11 \text{ cm} and d=3.5 cmd = 3.5 \text{ cm} into the volume expression.

    V=112π(3.5)2(3×11+3.5)V = \frac{1}{12} \pi (3.5)^2 (3 \times 11 + 3.5)

  2. Calculate the numerical value, leaving the answer in terms of π\pi.

    V=112π(12.25)(33+3.5)V = \frac{1}{12} \pi (12.25) (33 + 3.5) V=112π(12.25)(36.5)V = \frac{1}{12} \pi (12.25) (36.5) V=112π(447.125)V = \frac{1}{12} \pi (447.125)

    To express this as a fraction:

    V=447.12512π=357796πV = \frac{447.125}{12} \pi = \frac{3577}{96} \pi

    The internal volume of the salt shaker is:

    V=357796π cm3\boxed{V = \frac{3577}{96} \pi \text{ cm}^3}
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