Hemisphere Volume Calculation Explained

The question asks us to find the volume of a hemisphere given its diameter.

Understanding the Formula

First, we need the formula for the volume of a hemisphere. A hemisphere is half of a sphere.

• The volume of a full sphere is given by the formula: $V_{\text{sphere}} = \frac{4}{3} \pi r^3$, where $r$ is the radius.
• Therefore, the volume of a hemisphere is half of the sphere's volume: $V_{\text{hemisphere}} = \frac{1}{2} \times V_{\text{sphere}} = \frac{1}{2} \times \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3$.

Calculating the Radius

The problem gives the diameter of the hemisphere, which is 14 cm.

• The radius ($r$) is half of the diameter.
• Calculation: $r = \frac{\text{Diameter}}{2} = \frac{14 \text{ cm}}{2} = 7 \text{ cm}$.

Calculating the Volume

Now we can substitute the radius into the hemisphere volume formula. We'll use the approximation $\pi \approx \frac{22}{7}$ for calculation, as it simplifies nicely with a radius of 7 cm.

• Formula: $V_{\text{hemisphere}} = \frac{2}{3} \pi r^3$
• Substitute $r = 7$ cm and $\pi \approx \frac{22}{7}$: $V_{\text{hemisphere}} = \frac{2}{3} \times \frac{22}{7} \times (7 \text{ cm})^3$
• Expand $(7 \text{ cm})^3$: $V_{\text{hemisphere}} = \frac{2}{3} \times \frac{22}{7} \times (7 \times 7 \times 7) \text{ cm}^3$
• Cancel out one 7 from the numerator and denominator: $V_{\text{hemisphere}} = \frac{2}{3} \times 22 \times (7 \times 7) \text{ cm}^3$
• Calculate $7 \times 7$: $V_{\text{hemisphere}} = \frac{2}{3} \times 22 \times 49 \text{ cm}^3$
• Multiply the numbers in the numerator: $V_{\text{hemisphere}} = \frac{2 \times 22 \times 49}{3} \text{ cm}^3 = \frac{44 \times 49}{3} \text{ cm}^3$
• Calculate $44 \times 49$: $44 \times 49 = 44 \times (50 - 1) = 2200 - 44 = 2156$
• So, the volume is: $V_{\text{hemisphere}} = \frac{2156}{3} \text{ cm}^3$
• Convert the fraction to a decimal: $V_{\text{hemisphere}} \approx 718.666... \text{ cm}^3$
• Rounding to two decimal places, we get $718.67 \text{ cm}^3$.

Conclusion

The calculated volume of the hemisphere with a diameter of 14 cm is approximately $718.67 \text{ cm}^3$. This matches one of the options provided.