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. Tell it in THREE sentences
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Let's assume the height and diameter of the cone are both 'x'. Therefore, the volume of the cone can be calculated using the formula V = (1/3)πr^2h and the volume of the sphere can be calculated using the formula V = (4/3)πr^3. Since the diameter of the cone is x, the radius of the cone is x/2 and the radius of the hemisphere is also x/2. By equating the volumes of the cone and the hemisphere, we can solve for x and find the total volume of ice cream in and on top of the cone.
So what is the volumr
To find the volume, we need to first solve for the height of the cone.
Since the diameter of the cone is equal to its height (which we assumed as 'x'), the radius of the cone is x/2.
Setting the volume of the cone equal to the volume of the hemisphere, we have:
(1/3)π(x/2)^2(x) = (4/3)π(x/2)^3
Simplifying the equation, we get:
(x/4)(x) = (1/8)(x)^3
Multiplying both sides by 4 and rearranging, we get:
x^2 = (1/2)(x)^3
Dividing both sides by x^2, we get:
1 = (1/2)x
Simplifying, we find:
x = 2 inches
Since the volume of the cone is (1/3)πr^2h, we can substitute the height of 2 inches and radius of 1 inch to find:
V = (1/3)π(1)^2(2)
V = (2/3)π
The volume of the cone is (2/3)π cubic inches.
Adding the volume of the hemisphere on top, which is 4 cubic inches, the total volume of ice cream in and on top of the cone is (2/3)π + 4 cubic inches.
Since the diameter of the cone is equal to its height (which we assumed as 'x'), the radius of the cone is x/2.
Setting the volume of the cone equal to the volume of the hemisphere, we have:
(1/3)π(x/2)^2(x) = (4/3)π(x/2)^3
Simplifying the equation, we get:
(x/4)(x) = (1/8)(x)^3
Multiplying both sides by 4 and rearranging, we get:
x^2 = (1/2)(x)^3
Dividing both sides by x^2, we get:
1 = (1/2)x
Simplifying, we find:
x = 2 inches
Since the volume of the cone is (1/3)πr^2h, we can substitute the height of 2 inches and radius of 1 inch to find:
V = (1/3)π(1)^2(2)
V = (2/3)π
The volume of the cone is (2/3)π cubic inches.
Adding the volume of the hemisphere on top, which is 4 cubic inches, the total volume of ice cream in and on top of the cone is (2/3)π + 4 cubic inches.
What’s the answer to the cone?
The volume of the cone is (2/3)π cubic inches.
1 answer