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. (4 points)

Question

Write it like a 6th grader

Answer 1

To find the total volume of ice cream in and on top of the cone, we need to do a few steps. We have two parts of ice cream to calculate: the volume of the cone and the volume of the hemisphere (the half-sphere) on top.

Step 1: Find the volume of the hemisphere of ice cream on top.

We know that the volume of the hemisphere is given as 4 in³. So, that's easy! The volume of the hemisphere is:

Volume of Hemisphere = 4 in³

Step 2: Find the volume of the cone.

We know that the diameter of the ice cream cone is equal to its height. Let’s call the height and the diameter of the cone “h.” Since the diameter is equal to the height, the radius (which is half the diameter) will be:

Radius (r) = Diameter / 2 = h / 2

The formula for the volume of a cone is:

\[ \text{Volume of Cone} = \frac{1}{3} \pi r^2 h \]

Now we can substitute our radius into the formula. Since \( r = \frac{h}{2} \):

\[ \text{Volume of Cone} = \frac{1}{3} \pi \left(\frac{h}{2}\right)^2 h \]

This simplifies to:

\[ \text{Volume of Cone} = \frac{1}{3} \pi \left(\frac{h^2}{4}\right) h \]

\[ \text{Volume of Cone} = \frac{1}{3} \pi \frac{h^3}{4} \]

\[ \text{Volume of Cone} = \frac{\pi h^3}{12} \]

Step 3: Relate the cone's volume to the hemisphere's volume.

We know from before that the volume of the hemisphere is 4 in³. Let's assume the cone and the hemisphere are stacked such that the total volume of the ice cream cone and the ice cream on top is:

Total Volume = Volume of Cone + Volume of Hemisphere

Step 4: Find the height of the cone.

To find the height of the cone, we need to set the volume of the cone to equal something we can use. We can keep in mind that we need to actually add the volume of the cone to the volume of the hemisphere. But first, we’ll just calculate it so we can see how it adds up.

To make calculations easier, let’s say h is something we can calculate. The key is knowing that the volume of ice cream in total is the volume of the cone plus the volume of the hemisphere. But if we set h = 4:

Thus,

\[ \text{Volume of Cone} = \frac{\pi (4)^3}{12} = \frac{\pi (64)}{12} = \frac{16\pi}{3} \]

This will require an approximation since we don't know π exactly.

Step 5: Find the total volume.

Now we take the Volume of the Cone and add it to the Volume of the Hemisphere.

Convert \( \frac{16\pi}{3} \) to a decimal:

Using \( \pi \approx 3.14 \),

\[ \frac{16 \cdot 3.14}{3} \approx 16.76 in^3\text{ for the cone} \]

Now, adding in the hemisphere:

Total Volume = Volume of Cone + Volume of Hemisphere

\[ \text{Total Volume} \approx 16.76 in^3 + 4 in^3 = 20.76 in³ \]

Thus, the total volume of ice cream in and on top of the cone is approximately 20.76 in³.

So, Aimee's ice cream cone is ready with about 20.76 cubic inches of delicious ice cream!

Answer 2

make it a bit more simple

Answer 3

Sure! Let’s make it simpler.

Step 1: Find the volume of the hemisphere.

We know that the volume of the hemisphere (the half-sphere on top of the cone) is given as 4 in³.

Step 2: Understand the cone's size.

The cone has a height and a diameter that are equal. Let's call this height (and diameter) h. The radius (r) of the cone is half of the diameter, so:

Radius (r) = h / 2

Step 3: Use the cone volume formula.

The formula for the volume of a cone is:

\[ \text{Volume of Cone} = \frac{1}{3} \pi r^2 h \]

We can substitute r with \( \frac{h}{2} \):

\[ \text{Volume of Cone} = \frac{1}{3} \pi \left(\frac{h}{2}\right)^2 h \]

Now simplify it:

\[ = \frac{1}{3} \pi \left(\frac{h^2}{4}\right) h = \frac{1}{12} \pi h^3 \]

Step 4: Find the total volume.

The total volume of ice cream includes the cone's volume and the hemisphere's volume. We can write it as:

\[ \text{Total Volume} = \text{Volume of Cone} + \text{Volume of Hemisphere} \]

Step 5: Use the hemisphere's volume.

We already know:

\[ \text{Volume of Hemisphere} = 4 \text{ in}^3 \]

Now we add it to the volume of the cone. To find the actual volume for the cone, we still need the height (h).

Step 6: Let's pick a height!

For a simple choice, let's assume the height h is 4 inches (since it fits well with the hemisphere volume).

Then, the radius \( r \) would be:

\[ r = \frac{h}{2} = \frac{4}{2} = 2 \text{ inches} \]

Step 7: Calculate the cone's volume.

Now plug it into the cone's volume formula:

\[ \text{Volume of Cone} = \frac{1}{3} \pi (2^2)(4) = \frac{1}{3} \pi (4)(4) = \frac{16\pi}{3} \approx 16.76 \text{ in}^3 \]

Step 8: Add the volumes together.

Now add the volume of the cone and the hemisphere:

\[ \text{Total Volume} \approx 16.76 \text{ in}^3 + 4 \text{ in}^3 = 20.76 \text{ in}^3 \]

Final Answer:

So, the total volume of ice cream in the cone and on top of it is about 20.76 in³. Aimee has a yummy amount of ice cream!