An ice cream cone is 5 inches high and has an opening 3 inches in diameter. It is filled with ice cream and then given a spherical top. How much ice cream is there? The top part is 1/2 of a sphere.
vol of half sphere = (1/2)(4/3)pi r^3
vol of cone = (1/3)(5)(pi r^2)
where r = 1.5
To determine the volume of ice cream in the cone, we need to calculate the volume of the cone and the volume of the spherical top separately, and then add them together.
Let's start by calculating the volume of the cone:
The formula for the volume of a cone is V = 1/3 * pi * r^2 * h, where V is the volume, r is the radius of the base, and h is the height of the cone.
Given that the cone has an opening diameter of 3 inches, the radius (r) can be calculated by dividing the diameter by 2:
r = 3 inches / 2 = 1.5 inches
The height (h) of the cone is given as 5 inches.
Now, we can calculate the volume of the cone:
V_cone = 1/3 * pi * (1.5 inches)^2 * 5 inches
Next, we need to calculate the volume of the spherical top:
The formula for the volume of a sphere is V = 4/3 * pi * r^3, where V is the volume, and r is the radius of the sphere.
Since the top is described as half a sphere, we need to calculate the volume of the full sphere and then divide it by 2:
V_spherical_top = (1/2) * 4/3 * pi * (1.5 inches)^3
Now, we can add the volume of the cone and the volume of the spherical top to find the total volume of ice cream:
Total volume = V_cone + V_spherical_top
By substituting the numerical values into the equations and performing the calculations, you will find the total volume of ice cream in the cone.
To find out how much ice cream is in the cone, we need to calculate the volume of the cone and the volume of the half-sphere on top.
First, let's calculate the volume of the cone using the formula: V_cone = (1/3) * π * r^2 * h, where r is the radius of the base and h is the height of the cone.
Given that the diameter of the opening is 3 inches, we can find the radius by dividing the diameter by 2.
radius = 3 inches / 2 = 1.5 inches
The height of the cone is given as 5 inches.
Substituting these values into the formula, we get:
V_cone = (1/3) * π * (1.5 inches)^2 * 5 inches
Next, let's calculate the volume of the half-sphere using the formula: V_spherical = (2/3) * π * r^3, where r is the radius of the sphere.
Since the sphere is formed by closing the opening of the cone, the radius will be the same as the radius of the opening, which is 1.5 inches.
Substituting this value into the formula, we get:
V_spherical = (1/2) * (2/3) * π * (1.5 inches)^3
Finally, to find the total amount of ice cream, we add the volumes of the cone and half-sphere together.
Total ice cream volume = V_cone + V_spherical
Now, you can substitute the values and calculate the final answer.