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Question

A hemisphere is placed on top of an inverted cone. The back of the circumference, the radius, and the perpendicular height of the cone are marked with dashed lines. The radius and the height make a right angle and are perpendicular.

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)

do it like a high schooler pls

Answer 1

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

Step 1: Calculate the volume of the hemisphere

We know that the volume of a sphere \( V \) is given by the formula:

\[ V = \frac{4}{3} \pi r^3 \]

Since Aimee puts a perfect hemisphere of ice cream on top, the volume of the hemisphere is half of the volume of the sphere:

\[ V_{hemisphere} = \frac{1}{2} \left( \frac{4}{3} \pi r^3 \right) = \frac{2}{3} \pi r^3 \]

We are given that the volume of the ice cream in the hemisphere is \( 4 , \text{in}^3 \):

\[ \frac{2}{3} \pi r^3 = 4 \]

Step 2: Solve for the radius \( r \) of the hemisphere

To find \( r^3 \), we can rearrange the equation:

\[ \pi r^3 = 6 \quad \text{(by multiplying both sides by 3)} \] \[ r^3 = \frac{6}{\pi} \]

Now, we take the cube root to find \( r \):

\[ r = \sqrt[3]{\frac{6}{\pi}} \]

Step 3: Calculate the volume of the cone

The problem states that the diameter of the ice cream cone is equal to its height. Let's denote the height of the cone as \( h \). Since the diameter equals the height, the radius \( R \) of the cone would be half of the height:

\[ R = \frac{h}{2} \]

The volume \( V \) of a cone is calculated using the formula:

\[ V_{cone} = \frac{1}{3} \pi R^2 h \]

Substituting \( R = \frac{h}{2} \):

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

Simplifying this:

\[ = \frac{1}{3} \pi \frac{h^2}{4} h = \frac{1}{12} \pi h^3 \]

Step 4: Relate the height \( h \) to the radius of the hemisphere

Since the cone's diameter and height are equal (and the height is \( h \)), we need to express \( h \) in terms of the radius \( r \). Given that \( r \) is half of the diameter, we have:

\[ d = 2r \quad \Rightarrow \quad h = d = 2r \]

Substituting \( h = 2r \) into the volume formula for the cone:

\[ V_{cone} = \frac{1}{12} \pi (2r)^3 = \frac{1}{12} \pi (8r^3) = \frac{2}{3} \pi r^3 \]

Step 5: Calculate the total volume

Now we combine the volumes of the cone and hemisphere:

\[ V_{total} = V_{cone} + V_{hemisphere} = \frac{2}{3} \pi r^3 + 4 \]

From Step 1, we know:

\[ \frac{2}{3} \pi r^3 = 4 \quad \Rightarrow \quad V_{cone} = 4 \] And:

\[ V_{hemisphere} = 4 \]

Thus:

\[ V_{total} = 4 + 4 = 8 , \text{in}^3 \]

Final Answer

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