A large waffle cone has a height of 7.25 inches and a diameter of 3.25 inches. Two scoops of sorbet are placed on the wide end of the cone. The scoops are spheres with a diameter of 3.25 inches. If all the sorbet melts into the cone, will the cone overflow? Show all of your work.
My answer is no but I'm having a hard time showing my work for it.
Any advice or a hint?
volume of cone = (1/3)π r^2 h
= (1/3)π (1.625)^2 (7.25) = ... cubic inches
volume of 2 balls
= 2(4/3)π r^3
= (8/3)π (1.625)^3 = ....
compare your two answers
I will, thank you
To determine whether the cone will overflow when the sorbet melts, we need to compare the volume of the cone to the volume of the sorbet scoops.
1. Calculate the volume of the cone:
The formula for the volume of a cone is V = (1/3) * π * r^2 * h, where r is the radius of the base and h is the height.
Given:
diameter of the cone = 3.25 inches (So the radius of the cone, r = 3.25/2 = 1.625 inches)
height of the cone = 7.25 inches
Plugging the values into the formula:
V_cone = (1/3) * π * (1.625)^2 * 7.25
≈ 20.75 cubic inches
2. Calculate the volume of each sorbet scoop:
The formula for the volume of a sphere is V = (4/3) * π * r^3, where r is the radius of the sphere.
Given:
diameter of the sorbet scoop = 3.25 inches (So the radius of the scoop, r = 3.25/2 = 1.625 inches)
Plugging the value into the formula:
V_scoop = (4/3) * π * (1.625)^3
≈ 21.06 cubic inches
3. Total volume of both sorbet scoops:
Since we have two scoops, we need to calculate the volume of each scoop and then multiply it by 2.
Total V_sorbet = 2 * V_scoop
= 2 * 21.06
= 42.12 cubic inches
4. Compare the volumes:
If V_sorbet is greater than V_cone, then the cone will overflow. Otherwise, it won't.
V_sorbet ≈ 42.12 cubic inches
V_cone ≈ 20.75 cubic inches
Since V_sorbet is greater than V_cone (42.12 > 20.75), the cone will overflow when the sorbet melts.
To determine whether the cone will overflow when the sorbet melts, we need to compare the volume of the cone to the volume of the sorbet.
1. Calculate the volume of the cone:
The volume of a cone can be calculated using the formula V = (π * r^2 * h) / 3, where r is the radius of the base and h is the height.
Given:
- Diameter of the cone's base = 3.25 inches
- Radius of the cone's base (r) = (3.25 inches) / 2 = 1.625 inches
- Height of the cone (h) = 7.25 inches
Substituting the values into the formula:
V_cone = (π * (1.625 inches)^2 * 7.25 inches) / 3
V_cone = 37.268 cubic inches (approximately)
2. Calculate the volume of each sorbet scoop:
The volume of a sphere can be calculated using the formula V = (4/3) * π * r^3, where r is the radius of the sphere.
Given:
- Diameter of the sorbet scoop = 3.25 inches
- Radius of the sorbet scoop (r) = (3.25 inches) / 2 = 1.625 inches
Substituting the value into the formula:
V_scoop = (4/3) * π * (1.625 inches)^3
V_scoop = 14.138 cubic inches (approximately)
3. Calculate the total volume of the sorbet:
Since we have two scoops, the total volume of the sorbet is:
Total V_sorbet = 2 * V_scoop
Total V_sorbet = 2 * 14.138 cubic inches
Total V_sorbet = 28.276 cubic inches (approximately)
4. Compare the volumes:
If the total volume of the sorbet (28.276 cubic inches) is less than the volume of the cone (37.268 cubic inches), then the cone will not overflow when the sorbet melts.
Therefore, based on the calculations, the cone will not overflow when the sorbet melts.