Question

A metronome is a device that ticks at adjustable intervals to help keep time when playing music. The metronome on Ryan’s piano has the basic shape below.
A square pyramid. The base is 9 centimeters by 9 centimeters. The triangular sides have a base of 9 centimeters and height of 20.5 centimeters. The pyramid has a height of 20 centimeters.

If the metronome was created out of a piece of solid wood, what is the minimum amount of solid wood that was used?
369 centimeters cubed
450 centimeters cubed
540 centimeters cubed
810 centimeters cubed

Answer 1

To determine the minimum amount of solid wood that was used to create the metronome (which has the shape of a square pyramid), we need to calculate the volume of the pyramid.

The formula for the volume \( V \) of a square pyramid is given by:

\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]

### Step 1: Calculate the base area

The base of the pyramid is square with side length \( 9 \) cm. Therefore, the area of the base \( A \) is:

\[
A = \text{side} \times \text{side} = 9 \, \text{cm} \times 9 \, \text{cm} = 81 \, \text{cm}^2
\]

### Step 2: Calculate the volume of the pyramid

The height of the pyramid is given as \( 20 \) cm. Now we can use the volume formula:

\[
V = \frac{1}{3} \times 81 \, \text{cm}^2 \times 20 \, \text{cm}
\]

\[
V = \frac{1}{3} \times 81 \times 20 = \frac{1620}{3} = 540 \, \text{cm}^3
\]

### Conclusion

The minimum amount of solid wood that was used to create the metronome is:

\[
\boxed{540 \, \text{centimeters cubed}}
\]