given: volume of cone = 4

Question

height of cone = 2r
volume of cone = (1/3)π r^2 (2r) = 2/3 π r^3 = 4
π r^3 = 6
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.
can you explain it better
total volume = (1/2)(4/3)πr^3 + (1/3)π(r^2)(2r)
= (2/3) π r^3 + (2/3)π r^3
= (4/3)(6) = 8 in^3

Answer 1

Let's break down the problem step by step to clarify the calculations and explain the reasoning behind each step.

Given Information:

1. Volume of Ice Cream in Cone = 4 in³
2. Height of Cone = 2r (meaning height = double the radius)
3. Volume of Cone Formula: \[ V = \frac{1}{3} \pi r^2 h \] Substituting the height in this case, we have: \[ V = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]

From the problem statement:

• The volume of the cone is given as 4 cubic inches: \[ \frac{2}{3} \pi r^3 = 4 \]

Solve for Radius:

1. Rearranging for \( r^3 \): \[ 2 \pi r^3 = 12 ; \text{(multiplying both sides by 3)} \] \[ \pi r^3 = 6 \] \[ r^3 = \frac{6}{\pi} \]

Volume of Hemisphere:

Next, we add the volume of the perfect hemisphere on top of the cone. The volume \( V \) of a hemisphere is given by the formula: \[ V = \frac{2}{3} \pi r^3 \]

Total Volume of Ice Cream:

1. The total volume of ice cream consists of the volume inside the cone and the volume of the hemisphere: \[ \text{Total Volume} = \text{Volume of Cone} + \text{Volume of Hemisphere} \]

Using the previous calculations, we can substitute \( \frac{2}{3} \pi r^3 \) for the cone and \( \frac{2}{3} \pi r^3 \) for the hemisphere: \[ \text{Total Volume} = \frac{2}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{4}{3} \pi r^3 \]

Substitute \( r^3 \):

From our earlier calculation: \[ r^3 = \frac{6}{\pi} \]

Substitute this into the total volume formula: \[ \text{Total Volume} = \frac{4}{3} \pi \left(\frac{6}{\pi}\right) \] \[ = \frac{4 \cdot 6}{3} = 8 \text{ in³} \]

Conclusion:

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

This step-by-step breakdown explains how we derived the volumes and arrived at the final total volume of ice cream.

Answer 2

please put it into a paragraph

Answer 3

Aimee packs ice cream into an ice cream cone with a given volume of 4 in³, where the height of the cone is twice its radius (height = 2r). Using the formula for the volume of a cone, \(V = \frac{1}{3} \pi r^2 h\), and substituting the height, we find that the volume of the cone can be expressed as \(V = \frac{2}{3} \pi r^3\). Setting this equal to the given volume of 4 in³, we solve for the radius \(r\) and find that \(r^3 = \frac{6}{\pi}\). Next, we calculate the volume of a perfect hemisphere of ice cream placed on top of the cone, which is given by the formula \(V = \frac{2}{3} \pi r^3\). The total volume of ice cream, combining both the cone and the hemisphere, is calculated as \(\frac{2}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{4}{3} \pi r^3\). Substituting \(r^3 = \frac{6}{\pi}\) into this equation, we find the total volume to be \(8\) in³. Thus, the total volume of ice cream in and on top of the cone is 8 in³.

Answer 4

can you make it sound as an 8th grader

Answer 5

Aimee packs ice cream into an ice cream cone that has a volume of 4 cubic inches. The cool thing about this cone is that its height is twice the size of its radius (which we call r). To find out how big the radius is, we use the formula for the volume of a cone, which is \(V = \frac{1}{3} \pi r^2 h\). By plugging in the height as \(2r\), we get that the volume of the cone is \(\frac{2}{3} \pi r^3\). Since we know the volume is 4 cubic inches, we can set up the equation \(\frac{2}{3} \pi r^3 = 4\), and after some math, we figure out that \(r^3\) is \(\frac{6}{\pi}\).

Next, we want to find the volume of a perfect hemisphere of ice cream that sits on top of the cone. The formula for the volume of a hemisphere is \(\frac{2}{3} \pi r^3\). To find the total amount of ice cream, we add the volume of the cone and the volume of the hemisphere together. So, we get \(\frac{2}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{4}{3} \pi r^3\). Substituting our earlier result for \(r^3\), we calculate \(r^3\) to find that the total volume comes out to be 8 cubic inches. Therefore, the total amount of ice cream Aimee has in the cone and on top of it is 8 cubic inches!