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A geyser tank has a volume of 0.31246 m3. If the tank is 1315 mm long, what is its diameter? Use pi = 3.142. Round to 3 decimal places.

Mathematics

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Answer

0.550 m\text{0.550 m}

172.77 cm\text{172.77 cm}

Step-by-step solution

10 steps

To solve these problems, we will use the given formulas and dimensions. The geyser is a cylinder, and its length (1315 mm) will be used as the height in the volume formula.

3.1.1

  1. Identify the given values and the formula.

    The volume of the geyser (VV) is 0.31246 m30.31246 \text{ m}^3.
    The height (hh) of the geyser (length of the tank from the diagram) is 1315 mm1315 \text{ mm}.
    We are given π=3.142\pi = 3.142.
    The formula for the volume of a cylinder is V=π×radius2×heightV = \pi \times \text{radius}^2 \times \text{height}.

  2. Convert the height to meters.

    h=1315 mm=1315×11000 m=1.315 mh = 1315 \text{ mm} = 1315 \times \frac{1}{1000} \text{ m} = 1.315 \text{ m}
  3. Substitute the known values into the volume formula and solve for the radius squared (r2r^2).

    0.31246 m3=3.142×r2×1.315 m0.31246 \text{ m}^3 = 3.142 \times r^2 \times 1.315 \text{ m} 0.31246=4.13093×r20.31246 = 4.13093 \times r^2 r2=0.312464.13093r^2 = \frac{0.31246}{4.13093} r2≈0.07563806 m2r^2 \approx 0.07563806 \text{ m}^2
  4. Calculate the radius (rr).

    r=0.07563806r = \sqrt{0.07563806} r≈0.27502374 mr \approx 0.27502374 \text{ m}
  5. Calculate the diameter (D=2rD = 2r).

    D=2×0.27502374 mD = 2 \times 0.27502374 \text{ m} D≈0.55004748 mD \approx 0.55004748 \text{ m}
  6. Round the diameter to three decimal places.

    D≈0.550 mD \approx 0.550 \text{ m}

    The diameter of the geyser is 0.550 m\boxed{\text{0.550 m}}.

    3.1.2

  7. Identify the given values and the formula.

    We will use the unrounded diameter from the previous calculation for accuracy: D≈0.55004748 mD \approx 0.55004748 \text{ m}.
    We are given π=3.142\pi = 3.142.
    The formula for circumference is Circumference =π×diameter= \pi \times \text{diameter}.

  8. Convert the diameter to centimeters.

    D=0.55004748 m×100 cm/mD = 0.55004748 \text{ m} \times 100 \text{ cm/m} D≈55.004748 cmD \approx 55.004748 \text{ cm}
  9. Calculate the circumference.

    Circumference=3.142×55.004748 cm\text{Circumference} = 3.142 \times 55.004748 \text{ cm} Circumference≈172.7748 cm\text{Circumference} \approx 172.7748 \text{ cm}
  10. Round the circumference to two decimal places.

    Circumference≈172.77 cm\text{Circumference} \approx 172.77 \text{ cm}

    The circumference of the geyser is 172.77 cm\boxed{\text{172.77 cm}}.

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