Problem 49 - Chemistry: Principles and Reactions

Question

A waterbed filled with water has the dimensions \(8.0 \mathrm{ft} \times 7.0 \mathrm{ft} \times\) \(0.75 \mathrm{ft}\). Taking the density of water to be \(1.00 \mathrm{~g} / \mathrm{cm}^{3}\), how many kilograms of water are required to fill the waterbed?

Step-by-Step Solution

Answer

Answer: Approximately 1,194.77 kg.

Step 1: Convert dimensions to centimeters

Since the dimensions of the waterbed are given in feet, we must first convert them to centimeters, since the density of water is given in grams per cubic centimeter. 1 foot = 30.48 centimeters. So, the dimensions in centimeters are: 8.0 ft × 30.48 cm/ft = 243.84 cm 7.0 ft × 30.48 cm/ft = 213.36 cm 0.75 ft × 30.48 cm/ft = 22.86 cm

Step 2: Calculate the volume of the waterbed

Now, let's calculate the volume of the waterbed by multiplying its length, width, and height. Volume = length × width × height = 243.84 cm × 213.36 cm × 22.86 cm

Step 3: Find the weight of the water

Now we can find the weight of the water needed to fill the waterbed using the given density of water (1.00 g/cm³). Weight = Volume × Density = 243.84 cm × 213.36 cm × 22.86 cm × 1.00 g/cm³

Step 4: Convert weight to kilograms

Finally, we need to convert the weight of the water from grams to kilograms. 1 kilogram = 1000 grams Weight in kilograms = Weight in grams / 1000 g/kg Now, let's perform the calculations: Volume = 243.84 cm × 213.36 cm × 22.86 cm = 1,194,772.97 cm³ (approximately) Weight = 1,194,772.97 cm³ × 1.00 g/cm³ = 1,194,772.97 g Weight in kilograms = 1,194,772.97 g / 1000 g/kg = 1,194.77 kg (approximately) So, approximately 1,194.77 kilograms of water are required to fill the waterbed.

Key Concepts

Volume Conversion

When calculating the amount of water needed to fill a waterbed, the first important step is converting the measurements from feet to a metric unit like centimeters. This is crucial since density is often given in units like grams per cubic centimeter. The conversion factor here is straightforward: 1 foot equals 30.48 centimeters. Using this conversion factor ensures proper calculations.
To convert each dimension:

• Multiply the length in feet by 30.48 to get the length in centimeters.
• This means the 8.0 feet becomes approximately 243.84 cm.
• Similarly, convert the other dimensions accordingly: 7.0 ft converts to 213.36 cm, and 0.75 ft becomes 22.86 cm.

Thoroughly understanding volume conversion between units is essential in various practical applications, as it impacts accuracy in further calculations. Students should practice these conversions as they are a fundamental part of the metric system.

Metric System

The metric system plays a vital role in scientific measurements due to its systematic approach of units. Known for its simplicity and ease of scalability, it's widely used in scientific disciplines. The unit of measure in the metric system shifts by powers of ten, making calculations like conversions easier.
For instance:

• Length is measured in meters, and derived units such as centimeters (cm) and millimeters (mm).
• Volume is often measured in liters or the derived unit, cubic centimeters (cm³).
• Weight is commonly measured in grams and kilograms.

In the context of calculating the water needed for the waterbed, the reliance on converting units to the metric system simplifies the problem, making it straightforward and consistent. Each step builds on the previous one without needing complex conversions, thus highlighting the practicality of the metric system for comprehensive calculations.

Water Density

Understanding the concept of water density is crucial when solving problems involving liquids. Water typically has a density of 1.00 g/cm³ at room temperature, which implies that every cubic centimeter of water weighs one gram. This property simplifies calculations, as there is a one-to-one ratio between volume in cubic centimeters and mass in grams.
Given the water density, to find the weight:

• Calculate the total volume in cubic centimeters by multiplying the converted dimensions of the waterbed.
• Multiply this volume by the density of water, resulting in the water's mass in grams.

For practical applications, like identifying how many kilograms of water are required, you simply convert the final mass from grams to kilograms. Given that 1 kilogram equals 1000 grams, this conversion step offers clarity. Always remember, when you know the volume and the density, determining mass becomes a straightforward calculation. Water density remains a constant parameter, easing such computations.