How many spheres of radius 1.5 cm can be cut out of a wooden cube of edge 9 cm ?

Options

A. 216
B. 81
C. 27
D. can't be determined

Answer

To find out how many spheres of radius 1.5 cm can be cut out of a wooden cube with an edge of 9 cm, we will follow these steps:

### Step 1: Calculate the Volume of the Cube
The formula for the volume of a cube is given by:
\[ V_{\text{cube}} = \text{side}^3 \]

Given that the edge of the cube is 9 cm:
\[ V_{\text{cube}} = 9^3 = 729 \, \text{cm}^3 \]

### Step 2: Calculate the Volume of One Sphere
The formula for the volume of a sphere is:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]

Given that the radius \( r = 1.5 \, \text{cm} \):
\[ V_{\text{sphere}} = \frac{4}{3} \pi (1.5)^3 \]

Calculating \( (1.5)^3 \):
\[ (1.5)^3 = 3.375 \]

Now substituting this value into the volume formula:
\[ V_{\text{sphere}} = \frac{4}{3} \pi (3.375) \]
Using \( \pi \approx \frac{22}{7} \):
\[ V_{\text{sphere}} = \frac{4}{3} \times \frac{22}{7} \times 3.375 \]

Calculating further:
\[ V_{\text{sphere}} = \frac{4 \times 22 \times 3.375}{3 \times 7} \]
\[ V_{\text{sphere}} = \frac{297.0}{21} \approx 14.14 \, \text{cm}^3 \]

### Step 3: Calculate the Number of Spheres
To find the number of spheres \( n \) that can be cut out from the cube, we use the formula:
\[ n = \frac{V_{\text{cube}}}{V_{\text{sphere}}} \]

Substituting the values we calculated:
\[ n = \frac{729}{14.14} \approx 51.54 \]

Since we cannot have a fraction of a sphere, we take the whole number:
\[ n = 51 \]

### Conclusion
The number of spheres of radius 1.5 cm that can be cut out of a wooden cube of edge 9 cm is **51**.

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