Problem 24 - Introductory Chemistry Atoms First

Question

(a) 1222 pounds \(/ 2.0 \mathrm{in} .=?\) (b) 1222 pounds \(/ 2.00 \mathrm{in} .=?\) (c) What do you get when you quadruple \(21.72 \mathrm{~cm} ?\)

Step-by-Step Solution

Answer

(a) \( 611\ \dfrac{\text{pounds}}{\text{in}} \) (b) \( 611.00\ \dfrac{\text{pounds}}{\text{in}} \) (c) \( 86.88\ \text{cm} \)

Step 1: (a)Dividing 1222 pounds by 2 inches

Divide 1222 pounds by 2 inches: \( \dfrac{1222\text{ pounds}}{2.0\text{ in}} \) Now, perform the division: \( \dfrac{1222}{2} = 611 \) The result is: \( 611\ \dfrac{\text{pounds}}{\text{in}} \)

Step 2: (b)Dividing 1222 pounds by 2.00 inches

Divide 1222 pounds by 2.00 inches: \( \dfrac{1222\text{ pounds}}{2.00\text{ in}} \) Now, perform the division with consideration for decimal points: \( \dfrac{1222}{2.00} = 611.00 \) Keeping the appropriate significant figures, the result is: \( 611.00\ \dfrac{\text{pounds}}{\text{in}} \)

Step 3: (c)Quadrupling 21.72 cm

Quadruple 21.72 cm by multiplying it by 4: \( 4\times21.72\text{ cm} \) Now, perform the multiplication: \( 4\times21.72 = 86.88 \) The result is: \( 86.88\ \text{cm} \)

Key Concepts

Unit Conversion

Unit conversion is an essential skill in science and math that allows us to convert one measurement unit into another. This is crucial when comparing measurements or performing calculations, as it ensures consistency.
For example, if you're calculating the pressure exerted by a weight in pounds over an area in square inches, you need to express both weight and area in the same unit system. But in some scenarios, you might need to convert inches to centimeters or pounds to kilograms.
To successfully convert units:

• Identify the conversion factor: a ratio that expresses how many of one unit equals another.
• Multiply or divide the original measurement by this conversion factor.

Utilizing conversion factors, like 2.54 cm per inch, will help ensure your calculations are accurate and meaningful.

Division in Measurements

Division in measurements involves splitting a quantity by another, determining how many times one fits into the other. It requires careful attention to significant figures, especially when the measurements have different degrees of precision. For instance:

• When dividing 1222 pounds by 2.0 inches, you result in a measurement of 611 pounds per inch, considering only two significant figures.
• However, dividing by 2.00 inches leads to a result of 611.00 pounds per inch due to additional precision provided by the extra zero in the original measurement.

Remember that the result should carry the same number of significant figures as the measurement with the least.
It's essential to convey this precision in scientific notation or by rounding appropriately.

Multiplication in Measurements

Multiplication in measurements, just like division, requires attention to significant figures as well. This is because the precision of the result can only be as accurate as the least precise measurement used in the calculation.
When you quadruple 21.72 cm, you multiply it by 4:

• This results in 86.88 cm, which maintains the four significant figures from the original measurement of 21.72 cm.

Unlike division, multiplication typically doesn’t involve unit conversion. It's about scaling a quantity, commonly seen in repetitive situations or when proportions are involved.
The product will always carry the same amount of significant figures as the original measurement with the lowest precision.

Precision and Accuracy in Calculations

Precision refers to how consistently you can measure something with given methods and tools, while accuracy refers to how close the measured value is to the true or standard value. In calculations, precision ensures that results are repeatable, whereas accuracy ensures validity against a true standard.

• When dealing with significant figures, a lack of precision doesn't necessarily mean the measurement isn't accurate. It simply reflects the measurement's resolution.
• Adapt the number of significant figures in your calculations to only reflect what your instruments or methods can reliably measure or calculate.

By understanding and applying these concepts, you ensure that your results are both trustworthy and meaningful in scientific and mathematical contexts.