Aimee packs ice cream into an ice cream cone. She then puts a perfect hemisphere of ice cream on top of the cone that has a diameter of 5 cm. What is the volume of the hemisphere (half of a sphere), rounded to the nearest hundredth? *Show all steps to earn full credit.

1 answer

To find the volume of the hemisphere, we can start by using the formula for the volume of a sphere and then divide it by 2 since a hemisphere is half of a sphere.

The formula for the volume \( V \) of a sphere is:

\[ V = \frac{4}{3} \pi r^3 \]

where \( r \) is the radius of the sphere.

  1. Determine the radius: The problem states that the diameter of the hemisphere is 5 cm. The radius \( r \) is half of the diameter.

\[ r = \frac{5 , \text{cm}}{2} = 2.5 , \text{cm} \]

  1. Calculate the volume of the sphere using the radius:

\[ V_{\text{sphere}} = \frac{4}{3} \pi (2.5)^3 \]

  1. Calculate \( (2.5)^3 \):

\[ (2.5)^3 = 2.5 \times 2.5 \times 2.5 = 15.625 \]

  1. Substitute \( (2.5)^3 \) back into the volume formula:

\[ V_{\text{sphere}} = \frac{4}{3} \pi (15.625) \]

  1. Now calculate \( \frac{4}{3} \times 15.625 \):

\[ \frac{4}{3} \times 15.625 = \frac{62.5}{3} \approx 20.8333 \]

  1. Now multiply by \( \pi \). Using \( \pi \approx 3.14159 \):

\[ V_{\text{sphere}} \approx 20.8333 \times 3.14159 \approx 65.497 \]

  1. Since we want the volume of the hemisphere, we divide the volume of the sphere by 2:

\[ V_{\text{hemisphere}} = \frac{65.497}{2} \approx 32.7485 \]

  1. Round the answer to the nearest hundredth:

\[ V_{\text{hemisphere}} \approx 32.75 , \text{cm}^3 \]

Thus, the volume of the hemisphere, rounded to the nearest hundredth, is \( \boxed{32.75} \) cm³.