Question

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." 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. in 3 sentences and like an 8th grader wrote it

Answer 1

To find the total volume of ice cream in the cone and on top, we first need to calculate the volume of the cone. We know that the cone's volume is equal to its height since its diameter is equal to its height, which means its height is 2r (where r is the radius) and the volume formula for a cone is $V=\frac{1}{3}\pi r^{2}h$; so if we let the height be $h=2r$, the volume equation becomes $V=\frac{2}{3}\pi r^3$. Since the volume of the hemisphere is given as 4 in³ (and it has a volume formula of $V=\frac{2}{3}\pi r^3$), we can set it equal to 4 to find $r$ and then use that to find the volume of the cone, then we just add the two volumes together for the total.

1. Volume of the hemisphere: $$V=\frac{2}{3}\pi r^3=4$$ Solving for $r$ gives $r^3=6/\pi$.

2. Volume of the cone: Using the same $r^3$, we put $r$ back into the cone volume formula: $$V_{cone}=\frac{2}{3}\pi r^3=4$$ Which means the total volume of the ice cream in the cone and on top is: $$V_{total}=V_{cone}+V_{hemisphere}=4+4=8{\text{ in}}^3.$$ So the total volume of the ice cream in the cone and on top is 8 in³.