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 4in³ 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 short answer
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To solve this problem, let's start by finding the measurements of the ice-cream cone.
Let's assume the radius of the cone is r and the height of the cone is h.
We're given that the diameter of the cone (which is equal to 2r) is equal to its height (h), so we can write:
2r = h
The volume of a cone is given by the formula Vcone = (1/3)πr²h, and the volume of a hemisphere is given by the formula Vhemisphere = (2/3)πr³.
We know that the volume of the hemisphere is 4in³, so we can write:
(2/3)πr³ = 4
Now, let's manipulate this equation to solve for r:
πr³ = (3/2) * 4
πr³ = 6
r³ = 6/π
r ≈ 1.29 (rounded to two decimal places)
Now, substitute the value of r into the equation 2r = h:
2(1.29) = h
h ≈ 2.58 (rounded to two decimal places)
So, the height of the cone is approximately 2.58 inches.
Now, let's find the volume of the ice-cream in the cone.
Vcone = (1/3)π(1.29)²(2.58)
Vcone ≈ 1.68 in³ (rounded to two decimal places)
Finally, let's find the total volume of ice-cream in and on top of the cone by adding the volume of the cone to the volume of the hemisphere:
Total Volume = Vcone + Vhemisphere
≈ 1.68 + 4
≈ 5.68 in³
Therefore, the total volume of ice-cream in and on top of the cone is approximately 5.68 in³.