To find the total volume of ice cream in and on top of the cone, we need to find the volume of the cone and the volume of the hemisphere separately, and then add them together.
First, let's find the volume of the cone using the formula for the volume of a cone: V = 1/3 * π * r^2 * h
We are given that the diameter of the ice cream cone is equal to its height, so let's call the radius of the cone r and the height h. Since the diameter is equal to the height, we have r = h.
We are also given that the volume of the hemisphere is 4 in³. Since the volume of a hemisphere is half the volume of a sphere, the volume of the hemisphere is (1/2) * (4/3) * π * r^3.
Now, since the diameter of the cone is equal to its height, the radius of the hemisphere is also equal to the radius of the cone, which is r.
So, the volume of the hemisphere is (1/2) * (4/3) * π * r^3 = (2/3) * π * r^3.
The total volume of ice cream in and on top of the cone is the sum of the volume of the cone and the volume of the hemisphere:
Total volume = Volume of cone + Volume of hemisphere
= (1/3) * π * r^2 * h + (2/3) * π * r^3.
Since we know that r = h, we can substitute r for h in the formula:
Total volume = (1/3) * π * r^2 * r + (2/3) * π * r^3
= (π/3) * (r^3 + 2r^3)
= (π/3) * 3r^3
= πr^3.
So, the total volume of ice cream in and on top of the cone is πr^3.
Answer: The total volume of ice cream in and on top of the cone is πr^3.
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.
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