Cones, Cylinders, & Spheres Unit Test
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Question
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.
To solve this problem, we need to find the volume of both the cone and the hemisphere and add them together.
First, let's calculate the volume of the cone. We know that the volume of a cone is given by the formula:
V_cone = (1/3)πr^2h
Where r is the radius of the base of the cone, and h is the height of the cone.
In this case, we are given that the diameter of the ice cream cone is equal to its height. Let's call this value d.
Since the diameter is equal to the height, the radius would be half of the diameter, so the radius of the cone would be r = d/2.
Now, we also need to find the height of the cone. We are given that the volume of the hemisphere is 4 in^3, and we know that the diameter of the hemisphere is equal to the diameter (and height) of the cone.
The volume of the hemisphere is given by the formula:
V_hemisphere = (2/3)πr^3
We know that the volume of the hemisphere is 4 in^3, so we can solve for r using the formula above and then use r to find the height of the cone.
4 = (2/3)π(r)^3
r^3 = (3/2) * (4/π)
r^3 = 6/π
r = (6/π)^(1/3)
Since the radius of the cone is half of the diameter, we have:
h = d/2
= r/2
Now, we can substitute these values into the formula for the volume of the cone:
V_cone = (1/3)πr^2h
= (1/3)π((6/π)^(1/3))^2(r/2)
= (1/3)π(6/π)(6/π)(1/π)^(1/3)r
= 4/π^(1/3)r
Next, let's find the volume of the hemisphere. We already know the volume is 4 in^3.
Next, let's find the total volume of ice cream by adding the volume of the cone and the hemisphere:
Total volume = Volume of cone + Volume of hemisphere
= 4/π^(1/3)r + 4 in^3
Therefore, the total volume of ice cream in and on top of the cone is 4/π^(1/3)r + 4 in^3.