Problem 16 - Chemistry The Molecular Science

Question

Suppose a room is \(18 \mathrm{~m}\) long, \(15 \mathrm{~m}\) wide, and the distance from floor to ceiling is \(2.9 \mathrm{~m}\). What is the room's volume in cubic meters? In cubic centimeters? In liters?

Step-by-Step Solution

Answer

The room's volume is 783 cubic meters, 783,000,000 cubic centimeters, or 783,000 liters.

Step 1: Understand the problem

We need to find the volume of a room using given dimensions in different units: cubic meters, cubic centimeters, and liters. The dimensions provided are the length, width, and height of the room.

Step 2: Calculate volume in cubic meters

The volume of a rectangular prism is calculated by multiplying its length, width, and height. Given the dimensions, the volume is calculated as follows: \[ \text{Volume in cubic meters} = 18 \times 15 \times 2.9 = 783 \text{ cubic meters} \]

Step 3: Convert cubic meters to cubic centimeters

There are 100 centimeters in a meter, so we cube this conversion factor to convert cubic meters to cubic centimeters. \[ 1 \text{ cubic meter} = 100^3 \text{ cubic centimeters} = 1,000,000 \text{ cubic centimeters} \] Therefore, \[ 783 \text{ cubic meters} = 783 \times 1,000,000 = 783,000,000 \text{ cubic centimeters} \]

Step 4: Convert cubic meters to liters

Since 1 cubic meter is equivalent to 1,000 liters, we can convert the volume from cubic meters to liters by multiplying by this conversion factor. \[ 783 \text{ cubic meters} = 783 \times 1,000 = 783,000 \text{ liters} \]

Key Concepts

Cubic Meters

When discussing volume, especially in large spaces like rooms, cubic meters provide a standard way to express this measurement. A cubic meter is essentially a cube that measures one meter on each side.
This unit is particularly useful in construction, architecture, and shipping industries. To calculate the volume of a space in cubic meters, we multiply the length, width, and height of the space.
In our exercise, the room is 18 meters long, 15 meters wide, and 2.9 meters high. The formula for the volume of a rectangular prism is:

• Volume = length × width × height

Plugging in the values, we have:

• Volume = 18 × 15 × 2.9 = 783 cubic meters

So, the room has a volume of 783 cubic meters.

Cubic Centimeters

Cubic centimeters are an excellent unit for measuring smaller volumes. This is particularly useful in science and engineering for measuring things like the volume of small containers or components.
To convert cubic meters into cubic centimeters, we need to understand the relationship between the two. Since there are 100 centimeters in a meter, we cube this figure for converting between these cubic measures.
This means that:

• 1 cubic meter = 100 × 100 × 100 = 1,000,000 cubic centimeters

For the room from our exercise, we convert 783 cubic meters to cubic centimeters by multiplying by this factor:

• 783 cubic meters = 783 × 1,000,000 = 783,000,000 cubic centimeters

Thus, the room's volume is 783 million cubic centimeters.

Liters

Liters are commonly used to measure liquid volumes, but they also provide a convenient way to express the volume of a space. Since 1 cubic meter is equivalent to 1,000 liters, converting cubic meters into liters is straightforward.
This conversion is practical for contexts like water storage or calculating fluid capacities. Applying this conversion to our room, we have:

• 783 cubic meters = 783 × 1,000 = 783,000 liters

This means the room could theoretically hold 783,000 liters of water, highlighting the large capacity of the space as measured in liters.