Question

Food An ice cream cone is 10 centimeters deep and has a diameter of 4 centimeters. A spherical scoop of ice cream that is 4 centimeters in diameter rests on top of the cone. If all the ice cream melts into the cone, will the cone overflow? Explain.

Short answer

The cone will not overflow when the ice cream melts.

Step-by-step solution

Calculate the Volume of the Cone

The formula for the volume of a cone is \( V_{cone} = \frac{1}{3} \pi r^2 h \), where \( r \) is the radius and \( h \) is the height. The cone's diameter is 4 cm, so the radius is \( r = 2 \) cm, and the height \( h = 10 \) cm. Plug these values into the formula for the volume of the cone: \( V_{cone} = \frac{1}{3} \pi (2)^2 (10) = \frac{40}{3} \pi \) cubic centimeters.

Calculate the Volume of the Ice Cream Scoop

The ice cream scoop is a sphere with a diameter of 4 cm, so its radius is \( r = 2 \) cm. The formula for the volume of a sphere is \( V_{sphere} = \frac{4}{3} \pi r^3 \). Calculate the volume: \( V_{sphere} = \frac{4}{3} \pi (2)^3 = \frac{32}{3} \pi \) cubic centimeters.

Determine Total Volume of Ice Cream in Cone

If the scoop of ice cream melts completely, the total volume of ice cream is the volume of the sphere, \( \frac{32}{3} \pi \), which tries to fit into the cone.

Compare Ice Cream Volume with Cone Volume

The accepted volume of ice cream (from the sphere) is \( \frac{32}{3} \pi \), and the capacity of the cone is \( \frac{40}{3} \pi \). Since \( \frac{32}{3} \pi < \frac{40}{3} \pi \), the volume of the sphere is less than the volume that the cone can hold.

Conclusion

Since the volume of the ice cream \( \frac{32}{3} \pi \) is less than the volume of the cone \( \frac{40}{3} \pi \), the cone will not overflow when the ice cream melts.

Key concepts

Geometry

Geometry is a branch of mathematics that studies the shape, size, and properties of spaces and objects. It provides essential tools for understanding and calculating the dimensions and volumes of different shapes, like cones and spheres. When dealing with shapes in geometry, you often calculate measures such as perimeter, area, and volume.

Volume is a key concept when dealing with three-dimensional (3D) shapes. It refers to the amount of space an object occupies. To find the volume, you need to apply specific formulas, which differ based on the geometric shape you are examining. For instance:

• The formula for finding the volume of a cone involves the radius, height, and the constant \(\pi\).
• The formula for a sphere's volume considers its radius and the constant \(\pi\).

Understanding these principles allows you to solve problems like determining if a melted scoop of ice cream will overflow from a cone.

Cone Volume

Calculating the volume of a cone involves a straightforward formula. The cone is a 3D shape that narrows smoothly from a flat base to a point. The formula used is:

\[ V_{cone} = \frac{1}{3} \pi r^2 h \]

Where:

• \( V_{cone} \) stands for the volume of the cone.
• \( r \) is the radius of the base of the cone.
• \( h \) is the height of the cone.
• \( \pi \) is a constant approximately equal to 3.14159.

For an ice cream cone with a diameter of 4 centimeters, the radius is half the diameter, which is 2 centimeters. Given the height as 10 centimeters, you would substitute these values into the formula:

\[ V_{cone} = \frac{1}{3} \pi (2)^2 (10) = \frac{40}{3} \pi \]\
Thus, the cone can hold \( \frac{40}{3} \pi \) cubic centimeters of ice cream. Understanding this process helps in determining how other measurements would fit within a cone.

Sphere Volume

The volume of a sphere is found using a formula that accounts for the radius and the constant \(\pi\). A sphere is perfectly round and symmetrical around its center, like a ball. The formula is:

\[ V_{sphere} = \frac{4}{3} \pi r^3 \]

Where:

• \( V_{sphere} \) is the volume of the sphere.
• \( r \) is the radius of the sphere.
• \( \pi \) is approximately 3.14159.

In the case of a spherical ice cream scoop with a diameter of 4 centimeters, the radius is half of that, equaling 2 centimeters. Plugging this radius into the formula gives you:

\[ V_{sphere} = \frac{4}{3} \pi (2)^3 = \frac{32}{3} \pi \] \
This result shows that the ice cream, before melting, occupies \( \frac{32}{3} \pi \) cubic centimeters. Knowing how to compute this helps you understand the sphere's capacity and compare it with other volumes, such as that of a cone.