Question

The dimensions of a rectangular box are given to be \(10.2 \mathrm{~cm} \times 43.7 \mathrm{~cm} \times 9.56 \times 10^{2} \mathrm{~mm}\). What is its volume in liters? Be careful! The units are not all the same.

Short answer

The volume of the rectangular box is approximately 44.322 liters.

Step-by-step solution

Convert all units to the same unit

First, we need to convert all dimensions to the same unit. In this case, let's convert everything to centimeters. Note that 1 cm = 10 mm, so we can convert the given 9.56 x 10^2 mm dimension to centimeters by dividing by 10. \(9.56 \times 10^{2} \mathrm{~mm} = 9.56 \times 10^{2} \frac{\mathrm{cm}}{10\mathrm{mm}} = 95.6 \mathrm{~cm}\) Now, all three dimensions are in centimeters: Length = 10.2 cm Width = 43.7 cm Height = 95.6 cm

Calculate the volume of the box

Now that all of the dimensions are in centimeters, we can find the volume of the rectangular box using the formula for the volume of a rectangular box (Volume = Length × Width × Height): Volume = 10.2 cm × 43.7 cm × 95.6 cm To find the volume, multiply all three dimensions: Volume = 44322.384 cm³

Convert the volume to liters

Now, we need to convert the volume from cubic centimeters (cm³) to liters. We know that 1 liter = 1,000 cm³. To convert the volume to liters, we will divide the volume in cm³ by 1,000: Volume = 44322.384 cm³ × \(\frac{1 \mathrm{~L}}{1000 \mathrm{~cm^3}}\) Volume = 44.322384 liters

Provide the final answer

The volume of the rectangular box is approximately 44.322 liters.

Key concepts

Unit Conversion

Unit conversion is a fundamental skill in many fields, especially when dealing with dimensions or measurements. In this exercise, unit conversion involves transforming the dimensions of the rectangular box into the same unit to simplify calculations. To achieve this, it's crucial to identify the units involved and determine the conversion factors.

For instance, in the given problem, we had measurements in both centimeters and millimeters. Knowing the conversion factor between these two units is key: 1 centimeter equals 10 millimeters.

• When converting millimeters to centimeters, simply divide by 10.
• Conversely, when converting centimeters to millimeters, multiply by 10.

By doing this, calculations become straightforward, as all dimensions use the same unit, making further mathematical operations cohesive and error-free.

Rectangular Prism

A rectangular prism is a three-dimensional shape that has six faces, typically rectangular, with opposite faces being equal. These types of geometric figures are common in mathematics, representing various everyday objects like boxes or books.

To determine the volume of a rectangular prism, you need to understand its dimensions: length, width, and height. In a rectangular prism, the volume is calculated by multiplying these three dimensions together. The formula is as follows: \[ \text{Volume} = \text{Length} \times \text{Width} \times \text{Height} \] In this problem, after converting the dimensions to centimeters, you would compute the volume using this formula to get the result in cubic centimeters.

Cubic Centimeters to Liters

Converting from cubic centimeters to liters is an essential part of this exercise, especially given that many volume measurements in everyday scenarios use liters.

It's important to know that 1 liter is equivalent to 1,000 cubic centimeters (cm³). Therefore, the conversion from cm³ to liters is quite straightforward:

• Divide the volume in cubic centimeters by 1,000 to get the volume in liters.

In our example, after determining the volume of the prism as 44,322.384 cm³, we perform this conversion: \[ \text{Volume in liters} = \frac{44322.384}{1000} = 44.322384 \text{ liters} \] This conversion allows the resulting measurement to be better suited for practical uses.

Dimensional Analysis

Dimensional analysis involves analyzing the dimensions of the various units involved in a problem to ensure consistency and accuracy. It's a mathematical tool used for converting units and solving physical chemistry problems among other applications.

In the context of this exercise, dimensional analysis is employed both when arranging the formula for volume calculation and while converting the units from cubic centimeters to liters.

• This method ensures that the final units are correct and consistent with the target unit — liters in this case.
• By treating units algebraically, one can ensure smooth conversions and calculations, leading to accurate results.

Using dimensional analysis helps not only in keeping track of units during calculations but also in verifying the logic of the steps taken, thus ensuring reliability throughout the process.