The image depicts a cone with a hemisphere on top. The cone is drawn in beige, with a black outline and a dashed line down the center. The hemisphere is drawn in purple and white, with a black outline. The base of the hemisphere is a circle with a yellow outline and a black dashed line as its diameter. There is a black dot in the center of the circle where the diameter intersects the cone's dashed line. A right angle is drawn at the point where the diameter intersects the cone's dashed line. The image is a simple illustration of a cone with a hemisphere on top, and it is likely meant tAimee 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 volume of 4 in.3

Question

The diameter of the ice cream cone is equal to its height. What is the total volume of ice cream in and on top of the cone? Use the relationship between the formulas for the volumes of cones and spheres to help solve this problem. Show your work and explain your reasoning.o be used for teaching or learning about geometric shapes.

Answer 1

To solve the problem of finding the total volume of the ice cream in the cone and the hemisphere on top, we will need to use the volume formulas for a cone and a hemisphere.

1. Volume of the Hemisphere

We know the volume of the hemisphere is given as \( V_{hemisphere} = 4 \) in³.

2. Volume of the Cone

The formula for the volume of a cone is: \[ V_{cone} = \frac{1}{3} \pi r^2 h \] where \( r \) is the radius of the base of the cone and \( h \) is the height of the cone.

Given that the diameter of the cone is equal to its height, we have: \[ d = h \] Since the radius is half of the diameter, we can express the radius in terms of the height: \[ r = \frac{h}{2} \]

3. Substitute the Radius into the Volume Formula

Substituting \( r \) into the volume formula for the cone gives: \[ V_{cone} = \frac{1}{3} \pi \left( \frac{h}{2} \right)^2 h \] Calculating \( \left( \frac{h}{2} \right)^2 \): \[ V_{cone} = \frac{1}{3} \pi \left( \frac{h^2}{4} \right) h = \frac{1}{3} \pi \frac{h^3}{4} = \frac{\pi h^3}{12} \]

4. Equate the Volume of the Hemisphere and Find Height

Now, we will relate the volume of the hemisphere to the cone. Since we know the volume of the hemisphere is \( 4 \) in³, we can solve for \( h \):

1. Find the radius of the hemisphere using the volume formula: \[ V_{hemisphere} = \frac{2}{3} \pi r^3 \] Since we know \( V_{hemisphere} = 4 \): \[ 4 = \frac{2}{3} \pi r^3 \implies r^3 = \frac{4 \times 3}{2 \pi} = \frac{12}{2\pi} = \frac{6}{\pi} \] Thus, \[ r = \left( \frac{6}{\pi} \right)^{1/3} \]

2. The radius \( r \) of the cone must equal the radius of the hemisphere, meaning: \[ \frac{h}{2} = \left( \frac{6}{\pi} \right)^{1/3} \] Therefore, \( h = 2 \left( \frac{6}{\pi} \right)^{1/3} \).

5. Calculate the Volume of the Cone

Now substituting \( h \) back into the volume formula for the cone: \[ V_{cone} = \frac{\pi}{12} \left( 2 \left( \frac{6}{\pi} \right)^{1/3} \right)^3 = \frac{\pi}{12} \cdot 8 \cdot \frac{6}{\pi} = \frac{48}{12} = 4 \text{ in}^3 \]

6. Total Volume

Now, we calculate the total volume of the ice cream: \[ V_{total} = V_{cone} + V_{hemisphere} = 4 \text{ in}^3 + 4 \text{ in}^3 = 8 \text{ in}^3 \]

Conclusion

The total volume of the ice cream in and on top of the cone is \[ \boxed{8} \text{ in}^3. \]