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 in 4 sentances
(4 points)
To find the volume of the cone, we can use the formula V = (1/3)πr^2h. Since the diameter of the cone is equal to its height, we can say that the radius (r) is equal to half of the diameter. Let's call the radius of the cone "x", so the height is also "x".
So, the volume of the cone is V = (1/3)πx^2x.
To find the volume of the hemisphere, we use the formula V = (2/3)πr^3. Since the diameter of the hemisphere is equal to the diameter of the cone, the radius of the hemisphere is also "x".
So, the volume of the hemisphere is V = (2/3)πx^3.
The total volume is the sum of the volume of the cone and the volume of the hemisphere, so V_total = (1/3)πx^2x + (2/3)πx^3.
By substituting the given value of 4 in^3 for V_total and solving for x, we can find the value of x and then calculate the total volume of the ice cream.