hello i need a simplified solving answer to "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 volume of 4 in.3 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." the answer is 8 inch to 3

Answer 1

To solve this problem, we need to find the total volume of ice cream in and on top of the cone. We'll start with the information given and use the relationships between the volumes of cones and hemispheres (half-spheres).

1. Volume of the Hemisphere: The volume of the hemisphere of ice cream on top of the cone is given as \(4 , \text{in}^3\).

2. Volume of the Cone: We know that the ice cream cone's diameter is equal to its height. Let's denote the height of the cone as \(h\), and since the diameter equals the height, the radius \(r\) of the cone will be half of the height: \[ r = \frac{h}{2} \]

   The volume \(V\) of a cone is given by the formula: \[ V = \frac{1}{3} \pi r^2 h \]

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

3. Setting Volumes Equal: We need to find the height of the cone. Since the volume of the ice cream in the cone (which we just derived) and the volume of the hemisphere must add up to a total, we can set up the following equation: \[ \text{Volume of cone} + \text{Volume of hemisphere} = \text{Total volume} \] Given that the volume of the hemisphere is \(4 , \text{in}^3\), we have: \[ \frac{1}{12} \pi h^3 + 4 = \text{Total Volume} \]

4. Solving for Total Volume: We know from the problem that the total volume of the ice cream (in the cone and on top) is \(8 , \text{in}^3\). Therefore, we can set up this equation: \[ \frac{1}{12} \pi h^3 + 4 = 8 \] Solving for \(h\): \[ \frac{1}{12} \pi h^3 = 8 - 4 = 4 \] \[ \pi h^3 = 48 \] \[ h^3 = \frac{48}{\pi} \]

5. Calculating Volume of the Cone: Now we substitute \(h\) back into the formula for the volume of the cone: \[ V_{cone} = \frac{1}{12} \pi h^3 = \frac{1}{12} \pi \left(\frac{48}{\pi}\right) = \frac{48}{12} = 4 , \text{in}^3 \]

6. Total Volume: Finally, adding the two volumes: \[ \text{Total Volume} = V_{cone} + V_{hemisphere} = 4 + 4 = 8 , \text{in}^3 \]

Thus, the total volume of ice cream in and on top of the cone is \( \boxed{8} , \text{in}^3\).