Question

How many spheres of radius \(1.5 \mathrm{~cm}\) can be cut out of 2 wooden cube of edge \(9 \mathrm{~cm}\) ? (a) 216 (b) 81 (c) 27 (d) can't be determined

Short answer

a) 216 b) 81 c) 27 d) Can't be determined

Step-by-step solution

Calculate the volume of the sphere

The volume \(V_S\) of a sphere can be calculated using the following formula: \[ V_S = \frac{4}{3}\pi (r_S)^3 \] where \(r_S\) is the radius of the sphere. Substituting \(r_S = 1.5 \, \text{cm}\), the volume of the sphere is: \[ V_S = \frac{4}{3}\pi (1.5)^3 \] Solve this equation to get the volume of the sphere.

Calculate the volume of a single cube

The volume \(V_C\) of a cube is calculated using the following formula: \[ V_C = (s_C)^3 \] where \(s_C\) is the side length of the cube. Substituting \(s_C = 9 \, \text{cm}\), the volume of the cube is: \[ V_C = (9)^3 \] Solve this equation to get the volume of a cube.

Calculate the total volume of both cubes

We have two cubes of the same volume. So the total volume \(V_T\) is: \[ V_T = 2 \times V_C \] Substitute the value of \(V_C\) in the above equation to get the total volume.

Calculate the maximum number of spheres which fits in the cubes

To calculate how many spheres fit into our cubes, divide the total volume \(V_T\) by the volume of a single sphere \(V_S\). \[ n_S = \left\lfloor\frac{V_T}{V_S}\right\rfloor \] where \(n_S\) is the number of spheres, and the floor function \(\lfloor x \rfloor\) rounds \(x\) down to the nearest integer, because you can't have a fraction of a sphere. Solve this equation to determine the maximum number of spheres that can be cut out from the two cubes. The correct answer should then be matched with the given options (a) 216, (b) 81, (c) 27, or (d) can't be determined.

Key concepts

Volume Calculation

Volume calculation is all about determining how much space an object occupies.

It's a bit like trying to see how much water you could use to fill a container.

Understanding volume is crucial when trying to figure out how many smaller objects, such as spheres, can be fitted into a larger object, like a cube. You will often see volume calculations in terms of cubes, spheres, cylinders, and other three-dimensional shapes.

For example, in this problem, we start by finding the volume of both the sphere and the cube. The sphere volume tells us how much space each sphere takes, while the cube's volume shows us the total space available. By comparing these two volumes, we can figure out how many spheres fit inside the cubes.

This is why understanding how to calculate and compare volumes helps in determining how objects fit together.

Sphere Volume Formula

The sphere volume formula is used whenever you want to find out how much space a spherical object occupies.

The formula is given by:

• \[ V_S = \frac{4}{3}\pi r^3 \]

where:

• \(V_S\) is the volume of the sphere
• \(\pi\) is a constant, approximately 3.14159
• \(r\) is the radius of the sphere, which is half of its diameter

For example, if the radius of the sphere is 1.5 cm, then the volume would be:

• \( V_S = \frac{4}{3}\pi (1.5)^3 \)

This involves raising 1.5 to the third power (which means multiplying 1.5 by itself three times), and then multiplying by \(\pi\), factoring in the \(\frac{4}{3}\).

Using this formula helps to find the exact space a sphere occupies, which is essential when comparing with the cube space to determine how many spheres can fit.

Cube Volume Formula

To find the volume of a cube, we use the cube volume formula, which is straightforward but fundamental.

The formula is:

• \[ V_C = s^3 \]

Here,

• \(V_C\) is the volume of the cube,
• \(s\) is the length of one of the cube's sides.

A cube is unique because all its sides are equal. For example, if each edge of a cube is 9 cm long, the volume becomes:

• \( V_C = 9^3 \)

This means you multiply 9 by 9 by 9, resulting in the volume of one cube.

Calculating the cube's volume tells us how much space is inside the cube. When you have multiple cubes, you sum their volumes to see the total space available to fit other objects, like spheres. This is a key step for our sphere fitting problem.

Fitting Spheres into Cubes

The idea of fitting spheres into cubes involves determining how many spheres fit inside the given cube space.

Once you have the volumes, you use them to make this calculation. Here's the process:

• Start by calculating the volume for both the spheres and cubes, as previously explained.
• Next, calculate the total cube volume if there are multiple cubes. For instance, with two cubes, you'd double the volume of one cube.
• Divide the total cube volume by the volume of a single sphere. This gives you the maximum number of spheres that can fit.

However, there's a catch. Since spheres can't be cut into pieces fitting together seamlessly without gaps, we use the floor function.

This mathematical function rounds down to the nearest whole number, ensuring you consider only whole, complete spheres.

Therefore, this is how to calculate how many spheres can accurately fit into the space provided by the cubes, ensuring your answer is practical and feasible.