A spherical scoop of ice cream has radious 1.25 inches. The scoop is placed on top of a cone with radious of 1 inch and height of 5 inches. Is the cone large enough to hold all the ice cream if it melts? Show your calculations and explain your answer.
No; the volume of the ice cream is about
8.2 cubic inches, while the volume of the
cone is only about 5.2 cubic inches.
To determine if the cone is large enough to hold all the ice cream when it melts, we need to compare the volume of the ice cream scoop to the volume of the cone.
The volume of a sphere can be calculated using the formula:
V_sphere = (4/3) * π * r^3
Where r is the radius of the sphere.
The volume of a cone can be calculated using the formula:
V_cone = (1/3) * π * r^2 * h
Where r is the radius of the cone and h is the height of the cone.
Let's calculate the volumes of the ice cream scoop and the cone:
1. Volume of the ice cream scoop:
V_sphere = (4/3) * π * (1.25^3)
V_sphere ≈ 8.18 cubic inches (rounded to two decimal places)
2. Volume of the cone:
V_cone = (1/3) * π * (1^2) * 5
V_cone ≈ 5.24 cubic inches (rounded to two decimal places)
Comparing the volumes, we see that the volume of the ice cream scoop (8.18 cubic inches) is greater than the volume of the cone (5.24 cubic inches). Therefore, the cone is NOT large enough to hold all the ice cream when it melts.
Explanation:
The volume of the ice cream scoop was calculated using the formula for the volume of a sphere, which requires the radius of the sphere. The given radius of the ice cream scoop was 1.25 inches, so we substituted this value into the formula.
The volume of the cone was calculated using the formula for the volume of a cone, which requires the radius and height of the cone. The given radius of the cone was 1 inch and the given height was 5 inches, so we substituted these values into the formula.
By comparing the volumes, we determined that the ice cream scoop would not fit entirely within the cone when it melts.