To find the total volume of ice cream in and on top of the cone, we need to first calculate the volume of the cone. We know that the volume of the perfect hemisphere of ice cream on top is given as 4 cubic inches. The formula for the volume of a hemisphere is \(V = \frac{2}{3} \pi r^3\), and if we set this equal to 4, we can solve for \(r\) (the radius of the hemisphere). Rearranging gives us \(r^3 = 4 \times \frac{3}{2\pi} \approx 1.91\), so \(r \approx 1.23\) inches. Since the diameter of the hemisphere is 2r, the diameter is about 2.46 inches, meaning the cone also has a diameter of 2.46 inches, giving it a radius of approximately 1.23 inches. The height of the cone is equal to its diameter, so the height is also about 2.46 inches. Now we can calculate the volume of the cone using the formula \(V = \frac{1}{3} \pi r^2 h\). Plugging in our values gives \(V \approx \frac{1}{3} \pi (1.23^2)(2.46) \approx 3.08\) cubic inches. Now, we add the volume of the cone (approximately 3.08 in³) to the volume of the hemisphere (4 in³), which gives us a total volume of about \(3.08 + 4 = 7.08\) cubic inches of ice cream in and on top of the cone.
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
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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.
answer in one paragraph in an 8th grade level
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