To find the total volume of ice cream in and on top of the cone, I first determine the volume of the cone and the hemisphere of ice cream.
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Volume of the Hemisphere:
Aimee puts a hemisphere of ice cream on top of the cone with a volume of 4 cubic inches. This is already given. -
Volume of the Cone:
Since the cone's diameter is equal to its height, I can say that if the height \( h \) of the cone is \( d \), then the radius \( r \) is \( d/2 \) or \( h/2 \).The formula for the volume of a cone is:
\[ V = \frac{1}{3} \pi r^2 h \]Here, \( r = h/2 \). Substituting for \( r \) in the volume formula gives:
\[ V = \frac{1}{3} \pi \left(\frac{h}{2}\right)^2 h \] Simplifying this results in:
\[ V = \frac{1}{3} \pi \left(\frac{h^2}{4}\right) h = \frac{1}{12} \pi h^3 \] -
Equating Volumes:
We know that the volume of the hemisphere is 4 cubic inches. Therefore:
\[ \frac{1}{12} \pi h^3 = 4 \] -
Solving for \( h \):
Multiplying both sides by 12:
\[ \pi h^3 = 48 \] Dividing both sides by \( \pi \):
\[ h^3 = \frac{48}{\pi} \] Taking the cube root gives the height of the cone. -
Calculating the Volume of the Cone:
After determining \( h \), I would then substitute \( h \) back into the cone volume formula to find its volume. -
Total Volume:
Finally, I add the volume of the cone to the volume of the hemisphere to get the total volume of ice cream.Therefore, the total volume of ice cream in and on top of the cone is 8 cubic inches.