Problem 16 - Chemistry Matter and Change

Question

Challenge Calculate the areas and densities. Report the answers in the correct units. a. the area of a rectangle with sides measuring \(3 \times 10^{1} \mathrm{cm}\) and \(3 \times 10^{-2} \mathrm{cm}\) b. the area of a rectangle with sides measuring \(1 \times 10^{3} \mathrm{cm}\) and \(5 \times 10^{-1} \mathrm{cm}\) c. the density of a substance having a mass of \(9 \times 10^{5} \mathrm{g}\) and a volume of \(3 \times 10^{-1} \mathrm{cm}^{3}\) d. the density of a substance having a mass of \(4 \times 10^{-3} \mathrm{g}\) and a volume of \(2 \times 10^{-2} \mathrm{cm}^{3}\)

Step-by-Step Solution

Answer

a. 0.9 cm², b. 500 cm², c. \(3 \times 10^{6}\) g/cm³, d. 0.2 g/cm³.

Step 1: Calculate the Area for Part (a)

To find the area of a rectangle, multiply its length by its width. Here, the length is given as \(3 \times 10^{1} \) cm and the width is \(3 \times 10^{-2} \) cm. The area \(A\) is calculated as follows:\[ A = (3 \times 10^{1} \text{ cm}) \times (3 \times 10^{-2} \text{ cm}) = 9 \times 10^{-1} \text{ cm}^2 = 0.9 \text{ cm}^2 \]

Step 2: Calculate the Area for Part (b)

Similarly, calculate the area for the rectangle in part (b) using the given dimensions: \(1 \times 10^{3} \) cm and \(5 \times 10^{-1} \) cm.\[ A = (1 \times 10^{3} \text{ cm}) \times (5 \times 10^{-1} \text{ cm}) = 5 \times 10^{2} \text{ cm}^2 = 500 \text{ cm}^2 \]

Step 3: Calculate the Density for Part (c)

Density is calculated by dividing mass by volume. For part (c), the mass is \(9 \times 10^{5} \) g and the volume is \(3 \times 10^{-1} \) cm³. The density \(D\) is:\[ D = \frac{9 \times 10^{5} \text{ g}}{3 \times 10^{-1} \text{ cm}^3} = 3 \times 10^{6} \text{ g/cm}^3 \]

Step 4: Calculate the Density for Part (d)

For part (d), we use the given mass \(4 \times 10^{-3} \) g and volume \(2 \times 10^{-2} \) cm³ to find the density:\[ D = \frac{4 \times 10^{-3} \text{ g}}{2 \times 10^{-2} \text{ cm}^3} = 2 \times 10^{-1} \text{ g/cm}^3 = 0.2 \text{ g/cm}^3 \]

Key Concepts

Understanding Area Calculation

Calculating the area of a rectangle is one of the fundamental concepts in mathematics. The area of a rectangle is found by multiplying its length by its width. This gives the total number of square units that cover the rectangle.
For example, if we have a rectangle with lengths and widths expressed in scientific notation, like in the case of part (a) where the length is given as \(3 \times 10^{1}\) cm and the width as \(3 \times 10^{-2}\) cm, the multiplication follows the properties of exponents:

• Multiply the base numbers: \(3 \times 3 = 9\)
• Combine the powers of 10: \(10^{1-2} = 10^{-1}\)

So, the area becomes \(9 \times 10^{-1}\) cm², which is 0.9 cm² when expressed in standard notation.
This process simplifies complex-looking calculations and is very useful in problems that involve scientific notation. Remember to check your units, and ensure consistency for accurate results.

Mastering Scientific Notation

Scientific notation is a way to represent very large or very small numbers in a condensed form, which makes them easier to read and work with.
It involves the use of powers of 10, with a single non-zero digit to the left of the decimal point.

• For instance, the number 300 can be expressed as \(3 \times 10^{2}\)
• Likewise, the number 0.03 can be expressed as \(3 \times 10^{-2}\)

This method becomes crucial in calculations like those in our exercise, such as when multiplying side lengths for area calculations or dividing mass by volume for finding density.
Keeping track of the exponents assists in simplifying the arithmetic step by step. When multiplying terms with the same base, you add the exponents; when dividing, you subtract the exponents. This not only streamlines the process but also reduces errors.

The Importance of Unit Conversion

Unit conversion is the translation of a measure expressed in one set of units to another set of units without changing its value.
This step is often necessary when the initial figures are presented in a different form or when the problem's required answer is in another unit. In our exercise, although the units are consistent across each part (cm and g/cm³), it's crucial to ensure conversion accuracy when working with complex problems.
For instance, if results need to be given in different units, such as converting cm² to m², you should:

• Understand the conversion factor: 1 m = 100 cm, therefore 1 cm² = 0.0001 m²
• Apply appropriately calculated conversion factors to the values obtained.

This careful checking and conversion assure correct and meaningful communication of the results, aligning with the problem's specifications.