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.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) Put in simple explanation please and show the example in all words, no numbers

Answer 1

To find the total volume of ice cream both in and on top of the cone, we need to calculate the volume of the ice cream cone and the volume of the hemisphere on top of it.

Step 1: Volume of the Hemisphere

We know that the volume of the hemisphere is given to be a specific number. So, we start with that.

The formula for the volume of a hemisphere can be explained as half the volume of a full sphere. To find the volume of the hemisphere, we use the sphere formula and then divide that by two:

1. Find the volume of a full sphere using its radius.
2. Divide the sphere's volume by two to get the volume of the hemisphere.

Since it is given that the volume of the hemisphere of ice cream is already known, we will use that directly and say that it is a specific value which in this case is four cubic inches.

Step 2: Volume of the Cone

Next, we consider the volume of the cone. We know that the volume of a cone is calculated using the formula that includes both its radius and height. In this scenario, we are told that the diameter of the cone equals its height.

To relate this, we recognize that the diameter is twice the radius. So, if we let the diameter be "X," then the radius of the cone will be half of that, meaning it is "X/2."

Now, since the height is equal to the diameter (X), we can substitute diameter with height in the cone’s volume formula.

The formula for the volume of the cone involves multiplying the base area (which is based on the radius) with the height, and then dividing that product by three.

So, the steps would be to:

1. Calculate the base area using the radius we found.
2. Multiply that area by the height of the cone.
3. Divide the outcome by three to find the volume of the cone.

Step 3: Total Volume of Ice Cream

Finally, to find the total volume of ice cream, we add the volume of the hemisphere to the volume of the cone.

In summary, the total volume of ice cream is the sum of the volume that fills the cone and the volume represented by the hemisphere on top of the cone.

So, we add together:

• The known volume of the hemisphere (which is already given),
• And the calculated volume of the cone.

This gives us the total volume of ice cream in and on top of the cone.

In essence, the total volume represents all the ice cream in the cone and the smooth hemisphere on top, giving a full picture of how much ice cream Aimee has inside the cone and on top of it.