Question

By using estimation techniques, determine which of the following is the heaviest and which is the lightest: a 5-lb bag of potatoes, a \(5-\mathrm{kg}\) bag of sugar, or 1 gal of water (density \(=1.0 \mathrm{~g} / \mathrm{mL}\) ).

Short answer

The heaviest item is the 5-kg bag of sugar, and the lightest item is the 5-lb bag of potatoes.

Step-by-step solution

Convert 5 lb of potatoes to kilograms

To convert pounds to kilograms, we can use the conversion factor: 1 pound ≈ 0.4536 kilograms. So, 5 pounds of potatoes can be converted to kilograms as: 5 lb × (0.4536 kg / 1 lb) ≈ 2.27 kg

Convert 1 gal of water to liters

To convert gallons to liters, we can use the conversion factor: 1 gallon ≈ 3.7854 liters. So, 1 gallon of water can be converted to liters as: 1 gal × (3.7854 L / 1 gal) ≈ 3.79 L

Calculate the weight of 1 gal of water in kilograms

We know the density of water is 1.0 g/mL, and 1 L of water is equal to 1,000 mL. Given 3.79 L of water, we can find the weight in grams and then convert it to kilograms by dividing by 1,000. 3.79 L × (1,000 mL / 1 L) × (1.0 g / 1 mL) ≈ 3,790 g To convert grams to kilograms, we can use the conversion factor: 1,000 g ≈ 1 kg. So, 3,790 g of water can be converted to kilograms as: 3,790 g × (1 kg / 1,000 g) ≈ 3.79 kg

Compare the weights of the three items

Now that we have all the weights in kilograms, we can compare them to determine the heaviest and lightest items. 5-lb bag of potatoes: 2.27 kg 5-kg bag of sugar: 5 kg 1 gal of water: 3.79 kg

Identify the heaviest and lightest items

Comparing the weights, we can determine that the 5-kg bag of sugar (5 kg) is the heaviest, and the 5-lb bag of potatoes (2.27 kg) is the lightest. So, the heaviest item is the 5-kg bag of sugar, and the lightest item is the 5-lb bag of potatoes.

Key concepts

Unit Conversion

Understanding unit conversion is essential to compare measurements consistently in science, and particularly in chemistry. Converting units allows us to transform a quantity expressed in one system of measurement to another, using a conversion factor.

For instance, in the given exercise, to compare the weight of different items, we first need to convert their weights into a common unit. The conversion factor between pounds and kilograms is 1 lb \(\approx 0.4536\) kg. By multiplying the weight in pounds by this factor, we obtain its equivalent in kilograms.

This process was demonstrated when converting the 5 lb bag of potatoes to kilograms: \(5\) lb \(\times (0.4536\) kg / 1 lb) \(\approx 2.27\) kg. A similar process is applied to convert gallons to liters, another critical step in the given problem solving process.

Density Calculations

Density is a measure of how much mass is contained in a given volume and is crucial in identifying the quantity of matter present in a sample. The formula used to calculate density is: \[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\]

In the exercise, the density of water is given as 1.0 g/mL, which simplifies the process as the mass is equal to the volume in milliliters. To find the total weight of 1 gallon of water in kilograms, we must first convert the gallon into liters, then into milliliters, and finally use the density to find the mass.

This operation involves sequential conversions using the given density, starting with volume in liters and ending with mass in kilograms. Specifically, \(3.79\) L of water was converted to grams and then to kilograms. It is important to know that such conversions and calculations are not just academic exercises; they have practical applications in daily life and professional science fields.

Mass and Weight Comparison

In physics, mass refers to the quantity of matter in an object, while weight is the force exerted by gravity on that mass. However, often in everyday language and even in scientific contexts, 'weight' can be used to mean 'mass', especially when gravity is assumed to be constant, as is the case on the surface of the Earth.

In the final steps of the solution, we compared the mass of three different items after they were all converted to the same unit (kilograms). This comparison is straightforward once the conversions are made: you line up the numbers and see which is greatest and which is smallest. The 5-kg bag of sugar is the heaviest at 5 kg, and, conversely, the 5-lb bag of potatoes is the lightest at approximately 2.27 kg.

Comparing mass or weight can be useful in many real-world situations, such as shipping, where costs may be based on weight, or cooking, where recipe ingredients must be measured precisely. Understanding how to compare weights after converting different units is an essential skill in these contexts.