Problem 71 - Elementary Algebra

Question

Find the density of a piece of wood with a mass of \(12 \mathrm{~g}\) and a volume of \(15 \mathrm{~cm}^{3}\).

Step-by-Step Solution

Answer

The density is 0.8 g/cm³.

Step 1 - Understanding the formula for density

Density is calculated using the formula: \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]

Step 2 - Identify and substitute the given values

The mass of the wood is given as 12 grams, and the volume is given as 15 cubic centimeters. Substitute these values into the formula: \[ \text{Density} = \frac{12 \mathrm{~g}}{15 \mathrm{~cm}^{3}} \]

Step 3 - Perform the calculation

Perform the division to find the density: \[ \text{Density} = \frac{12}{15} \mathrm{~g/cm}^{3} = 0.8 \mathrm{~g/cm}^{3} \]

Key Concepts

Mass

Mass is a measure of the amount of matter in an object. When dealing with density problems, mass is typically given in grams (g). In our example, the mass of the piece of wood is given as 12 grams. To understand mass better: think of it as the weight but without the effects of gravity. It's an intrinsic property that doesn't change regardless of location.

A practical way to measure mass is using a scale. Always remember that in the context of density calculations, ensuring the mass is accurately measured is crucial for an accurate density result.

Volume

Volume is a measure of how much space an object occupies. It is typically measured in cubic centimeters (cm³) or liters (L). In the exercise, the volume of the piece of wood is given as 15 cm³. Volume can be a bit more abstract than mass, but it's essentially about filling space.

To measure the volume of a regular object, one can use mathematical formulas. For irregular objects, submerging them in water and measuring the water displacement is an effective method. Just like mass, accurate volume measurement is key to finding the correct density.

Density Formula

Density is defined as mass per unit volume. The formula to calculate density is:

\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
In our example, the mass is 12 grams, and the volume is 15 cubic centimeters. Plugging these values into the formula gives:

\[ \text{Density} = \frac{12 \mathrm{~g}}{15 \mathrm{~cm}^{3}} \]
By performing the division, we get:

\[ \text{Density} = 0.8 \mathrm{~g/cm}^{3} \]
This value means that for every cubic centimeter of the wood, it has a mass of 0.8 grams. Understanding and applying the density formula helps us compare different materials to see which is denser.