Problem 80 - Chemistry: An Atoms-Focused Approach

Question

Perform each of the following calculations, and express the answer with the correct number of significant figures: a. \([(12 \times 60.0)+55.3] /\left(5.000 \times 10^{3}\right)=\) b. \(3.1416 \times(2.031)^{2} \times 3.75 \times 8.00=\) c. The number of cubic centimeters in 389 cubic inches d. The average (mean) of \(8.7,8.5,8.5,8.9,\) and 8.8

Step-by-Step Solution

Answer

Answer: The answers for each problem are: a. 0.155 b. 393 c. 6370 cm³ d. 8.7

Step 1: Perform the calculation

Before we start, let's recall the correct order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Following the order of operations, we will first perform the multiplication: \((12\times60.0)=720\). Then, we will proceed with the addition: \(720+55.3=775.3\). Lastly, we will divide by the provided value: \(775.3/\left(5.000\times10^3\right)=775.3/5000=0.15506\).

Step 2: Express the answer with the correct number of significant figures

The least number of significant figures of numbers involved in the calculation is 3 (from 60.0). Therefore, the final answer should have 3 significant figures. Thus, we have: \(0.15506 \approx 0.155\). #Problem b: Calculate the given expression with correct significant figures#

Step 1: Perform the calculation

Following the order of operations, we will first calculate the exponent: \((2.031)^{2}=4.124961\). Then perform the multiplications: \(3.1416\times4.124961\times3.75\times8.00=392.603590816\).

Step 2: Express the answer with the correct number of significant figures

The least number of significant figures of numbers involved in the calculation is 3 (from 3.75 and 8.00). Therefore, the final answer should have 3 significant figures. Thus, we have: \(392.603590816 \approx 393\). #Problem c: Convert cubic inches to cubic centimeters#

Step 1: Conversion factor

1 cubic inch is equal to 16.3871 cubic centimeters. So, we have the conversion factor: \(1 \text{ in}^3 = 16.3871 \text{ cm}^3\).

Step 2: Perform the conversion

Multiply the given number of cubic inches (389) by the conversion factor: \(389 \text{ in}^3 \times 16.3871 \frac{\text{cm}^3}{\text{in}^3} = 6374.1229 \text{ cm}^3\).

Step 3: Express the answer with the correct number of significant figures

The given number of cubic inches (389) has 3 significant figures. Therefore, the final answer should have 3 significant figures. Thus, we have: \(6374.1229 \:text{ cm}^3 \approx 6370 \text{ cm}^3\). #Problem d: Find the mean of the numbers#

Step 1: Add the numbers

Add the given numbers: \(8.7+8.5+8.5+8.9+8.8=43.4\).

Step 2: Divide the sum by the number of terms

There are 5 numbers given in the set. Divide the sum by 5 to find the mean: \(43.4 \div 5 = 8.68\).

Step 3: Express the answer with the correct number of significant figures

The given numbers have 2 significant figures. Therefore, the final answer should have 2 significant figures. Thus, we have: \(8.68 \approx 8.7\). So, the answers for each problem are: a. \(0.155\) b. \(393\) c. \(6370 \text{ cm}^3\) d. \(8.7\)

Key Concepts

Order of Operations

Understanding the order of operations is crucial in mathematics to ensure consistent and correct results. This set of rules determines the sequence in which arithmetic operations should be carried out.

• First, solve any calculations inside parentheses.
• Next, evaluate exponents or powers.
• Then, perform multiplication and division, from left to right.
• Finally, work through addition and subtraction, also from left to right.

To remember these steps, many use the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This rule helps avoid errors in multi-step calculations and ensures the operation flows logically. By following PEMDAS, you can simplify intricate problems with confidence and accuracy. Ensure each answer reflects the original pre-set order of calculating.

Conversion Factors

Conversion factors are constants used to convert a quantity expressed in one unit to another. They are essentially ratios that equal one but allow the transformation of measurements while keeping the quantity's value the same.
For example, to convert cubic inches to cubic centimeters, we use the conversion factor of 1 cubic inch to 16.3871 cubic centimeters. This means every cubic inch is 16.3871 times larger when expressed in cubic centimeters.

• Identify the needed conversion factor.
• Multiply the quantity by this factor.
• Ensure that units cancel appropriately, leaving your desired unit intact.

This method allows for accurate unit conversions in scientific, engineering, and everyday measurements. It is useful to memorize certain common conversion factors, like the relationship between inches and centimeters, to streamline the calculation process.

Cubic Measurements

Cubic measurements denote volume, particularly when dealing with three-dimensional spaces or objects. These measurements are essential in fields like construction, shipping, and sciences concerned with physical space.
Common units include cubic inches, cubic centimeters, and cubic meters. When converting these units, it's crucial to apply the correct conversion factor, as seen with cubic inches to centimeters. Volume measurement ensures we can quantify how much space a material or object will occupy or require.
Understanding the context of measurement helps choose the correct unit. For smaller volumes, cubic centimeters are standard, while larger spaces might use cubic meters. Keeping track of unit consistency throughout calculations is crucial to avoid errors.

Mean Calculation

The mean, or average, is a measure of central tendency, which provides a central value for a set of numbers. Calculating a mean involves a few straightforward steps, but accuracy requires attention to detail, especially regarding significant figures.

• Add together all the numbers in your data set.
• Count the total number of values in the set.
• Divide the total sum by the number of values to get the mean.

For example, to find the mean of numbers like 8.7, 8.5, 8.5, 8.9, and 8.8, sum these values to get a total of 43.4. Dividing by 5 gives 8.68, which rounds to 8.7 with the appropriate significant figures. It's essential to report the mean with the right number of significant digits to reflect data precision.