Use the image to answer the 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)
Math Short Answer Rubric (4 points)
Points Awarded Criteria
4
The answer is clear and correct.
The answer shows that the student used problem-solving skills and reasoning.
All work is shown or explained.
3
The answer is mostly correct. The student made small errors while solving.
The answer shows that the student used some problem-solving skills and reasoning.
Most work is shown or explained.
2
The answer is partially correct. The student made errors while solving.
The answer shows that the student used some problem-solving skills or reasoning.
Some work is shown or explained.
1
The answer is incorrect. The student made errors while solving.
The answer shows little problem-solving skills or reasoning.
Little or no work is shown.
0 The question is not answered.
To find the total volume of ice cream in and on top of the cone, we need to find the volume of both the cone and the hemisphere and then add them together.
The volume of a cone is calculated using the formula V = 1/3 * π * r^2 * h, where r is the radius of the base and h is the height.
Since the diameter of the ice cream cone is equal to its height, we can let the radius of the base be r and the height be 2r.
The volume of the cone is then Vcone = 1/3 * π * r^2 * (2r) = 2/3 * π * r^3.
The volume of a hemisphere is calculated using the formula V = 2/3 * π * r^3.
Since the diameter of the hemisphere is equal to the diameter of the cone, the radius of the hemisphere is also r.
The volume of the hemisphere is then Vhemisphere = 2/3 * π * r^3.
The total volume of ice cream in and on top of the cone is V = Vcone + Vhemisphere.
Substituting in the values, we get V = 2/3 * π * r^3 + 2/3 * π * r^3 = 4/3 * π * r^3.
Since the volume of the hemisphere is given as 4 in.3, we can set up the equation 4/3 * π * r^3 = 4.
Dividing both sides of the equation by 4/3 * π, we get r^3 = 1.
Taking the cube root of both sides, we get r = 1.
So the radius of the cone and hemisphere is 1 inch.
Now we can substitute this value back into the equation V = 4/3 * π * r^3 to find the total volume.
V = 4/3 * π * (1^3) = 4/3 * π.
Therefore, the total volume of ice cream in and on top of the cone is 4/3 * π cubic inches.