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)
To find the total volume of ice cream in and on top of the cone, we need to calculate the volume of the cone and the volume of the hemisphere separately, and then add them together.
1. Volume of the cone:
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height.
In this case, the diameter of the ice cream cone is equal to its height, so the radius of the cone is half the diameter.
Let's assume the radius of the cone is r, then the height of the cone is also r.
Therefore, the volume of the cone is V_cone = (1/3)πr^2(r) = (1/3)πr^3.
2. Volume of the hemisphere:
The volume of a hemisphere is given by the formula V = (2/3)πr^3, where r is the radius.
In this case, the diameter of the hemisphere is also equal to the height of the cone, so the radius of the hemisphere is r.
Therefore, the volume of the hemisphere is V_hemisphere = (2/3)πr^3.
3. Total volume of ice cream in and on top of the cone:
The total volume is the sum of the volume of the cone and the volume of the hemisphere:
V_total = V_cone + V_hemisphere
= (1/3)πr^3 + (2/3)πr^3
= πr^3(1/3 + 2/3)
= πr^3.
We know that the volume of the ice cream in and on top of the cone is 4 in^3.
Therefore, we have the equation πr^3 = 4.
To solve for r, we can divide both sides of the equation by π and then take the cube root:
r^3 = 4/π
r = (4/π)^(1/3)
Finally, to find the total volume of ice cream in and on top of the cone, we substitute the value of r into the equation for V_total:
V_total = π(r^3)
= π((4/π)^(1/3))^3
= 4 in^3.
Therefore, the total volume of ice cream in and on top of the cone is 4 in^3.
can u shorten that
Sure!
To find the total volume of ice cream in and on top of the cone:
1. The volume of the cone is (1/3)πr^3.
2. The volume of the hemisphere is (2/3)πr^3.
3. The total volume is the sum of the volume of the cone and the volume of the hemisphere.
4. Let's assume the radius of the cone is r. The diameter of the cone is the same as its height.
5. The equation πr^3 = 4 gives us the value of r.
6. Substituting the value of r into the equation for the total volume, we get V_total = 4 in^3.
Therefore, the total volume of ice cream in and on top of the cone is 4 in^3.