Question

What is the difference between the amount of water that can be held by a sphere of diameter 1 centimeter and a cube 1 centimeter on a side? Use 3.14 for \(\pi\). A. \(0.477 \mathrm{~cm}^3\) B. \(0.523 \mathrm{~cm}^3\) C. \(4.770 \mathrm{~cm}^3\) D. \(5.230 \mathrm{~cm}^3\)

Short answer

0.477 \mathrm{cm}^3

Step-by-step solution

Understand the Problem

Determine the difference in volume between a sphere with a diameter of 1 centimeter and a cube with a side length of 1 centimeter.

Calculate the Volume of the Sphere

The volume of a sphere is given by the formula \( V_{sphere} = \frac{4}{3} \pi r^3 \). Here, the diameter is 1 centimeter, so the radius \( r \) is \( \frac{1}{2} = 0.5 \) cm. Plugging in the radius and \( \pi = 3.14 \), we get: \[ V_{sphere} = \frac{4}{3} \pi (0.5)^3 = \frac{4}{3} \cdot 3.14 \cdot 0.125 = \frac{4}{3} \cdot 3.14 \cdot 0.125 = 0.523 \ \mathrm{cm}^3 \]

Calculate the Volume of the Cube

The volume of a cube is given by the formula \ V_{cube} = s^3 \. Here, \( s \) (side length) is 1 centimeter. Therefore, the volume is: \[ V_{cube} = 1^3 = 1 \ \mathrm{cm}^3 \]

Find the Difference in Volumes

Subtract the volume of the sphere from the volume of the cube: \[ V_{difference} = V_{cube} - V_{sphere} = 1 - 0.523 = 0.477 \ \mathrm{cm}^3. \]

Choose the Correct Answer

From the calculated difference, \( 0.477 \mathrm{cm}^3 \), the correct answer is option A.

Key concepts

Understanding Sphere Volume

A sphere is a perfectly round three-dimensional shape. To find the volume of a sphere, you use the formula: \( V_{sphere} = \frac{4}{3} \pi r^3 \). This formula calculates how much space inside the sphere. Here,

• \( \pi \) is approximately 3.14 (a constant)
• \( r \) is the radius of the sphere

The radius is half of the diameter. If the diameter is 1 centimeter, the radius is 0.5 centimeters. Plug these values into the formula to get the volume.
Using our example:
\( V_{sphere} = \frac{4}{3} \times 3.14 \times (0.5)^3 = \frac{4}{3} \times 3.14 \times 0.125 = 0.523 \, \mathrm{cm}^3 \).
So, our sphere with a 1-centimeter diameter has a volume of 0.523 cubic centimeters.

Understanding Cube Volume

A cube is a three-dimensional shape with six equal squares as its faces. To calculate the volume of a cube, you use the formula: \( V_{cube} = s^3 \). This formula tells you how much space is inside the cube. Here,

• \( s \) is the length of a side of the cube.

If each side of the cube is 1 centimeter, then: \( V_{cube} = 1^3 = 1 \ \, \mathrm{cm}^3 \).
So, a cube with side lengths of 1 centimeter has a volume of 1 cubic centimeter.

Comparing Geometric Shapes: Sphere vs. Cube

When dealing with geometric shapes, it helps to compare their volumes to understand their capacity. In this exercise, we looked at a sphere and a cube, each with specific dimensions.

• For the sphere, we calculated a volume of 0.523 cubic centimeters.
• For the cube, we found the volume to be 1 cubic centimeter.

To find the difference, subtract the volume of the sphere from the volume of the cube: \( 1 \ \mathrm{cm}^3 - 0.523 \ \, \mathrm{cm}^3 = 0.477 \ \, \mathrm{cm}^3 \).
So, the difference in volume is 0.477 cubic centimeters, meaning the cube can hold this much more water than the sphere with a 1-centimeter diameter.