Question

The white blood cell concentration in normal blood is approximately 12,000 cells \(/ \mathrm{mm}^{3}\) of blood. How many white blood cells does a normal adult with \(5 \mathrm{~L}\) of blood have? Express the answer in scientific notation.

Short answer

The normal adult has approximately \(6.0 \times 10^{10}\) white blood cells.

Step-by-step solution

Understand the Problem

We need to find out the total number of white blood cells in an adult with 5 liters of blood, given the concentration is 12,000 white blood cells per cubic millimeter (mm i superscript{3}) of blood.

Convert Liters to Cubic Millimeters

First, convert the volume of blood from liters to cubic millimeters, because the concentration is given in cells per cubic millimeter. We know that 1 liter = 1,000,000 cubic millimeters. Therefore, 5 liters of blood equals:\[5 \text{ liters} \times 1,000,000 \text{ mm}^3/\text{liter} = 5,000,000 \text{ mm}^3\]

Calculate Total White Blood Cells

Multiply the concentration of white blood cells by the total volume of blood in cubic millimeters to find the total number of white blood cells:\[12,000 \text{ cells/mm}^3 \times 5,000,000 \text{ mm}^3 = 60,000,000,000 \text{ cells}\]

Express in Scientific Notation

Express the total number of white blood cells in scientific notation. The number is 60,000,000,000 cells, which can be written in scientific notation as:\[6.0 \times 10^{10} \text{ cells}\]

Key concepts

Scientific Notation

Scientific notation is a mathematical way to represent very large or very small numbers in a simplified manner. It is particularly useful in fields like chemistry and physics, where you often deal with numbers that can be unwieldy.

• A number in scientific notation is expressed as the product of two parts: a coefficient and a power of 10.
• The general form is: \[a \times 10^n\] where \(a\) is a number greater than or equal to 1 and less than 10, and \(n\) is an integer.

Scientific notation allows for easier comparison and calculation with these numbers. For example, the number 60,000,000,000 can be expressed as \(6.0 \times 10^{10}\). Here, \(6.0\) is the coefficient, and \(10^{10}\) represents the number of zeros following the 6. Writing numbers this way makes it simpler to handle huge values, particularly when performing arithmetic operations.

Liter to Cubic Millimeter Conversion

To solve many scientific and medical problems, it is crucial to understand how to convert measurements between different units. In this case, we are converting liters to cubic millimeters because the concentration of white blood cells is given per cubic millimeter.

• 1 liter is equivalent to 1,000,000 cubic millimeters.
• This conversion factor is important when you need to switch between volumes of liquid expressed in liters and the much smaller unit, cubic millimeters.

For our problem, we have 5 liters of blood. By applying the conversion factor, we find:\[5 \text{ liters} \times 1,000,000 \text{ mm}^3/\text{liter} = 5,000,000 \text{ mm}^3\]This calculation tells us exactly how much volume we are dealing with in terms of cubic millimeters, thereby aligning with the units in the concentration given.

Total White Blood Cells Calculation

Once you have both the concentration of cells and the total volume in compatible units, you can easily calculate the total number of cells. In this exercise, the concentration of white blood cells is given as 12,000 per cubic millimeter.

• The total number of white blood cells is calculated by multiplying the concentration by the total volume of blood in cubic millimeters.
• The formula for the calculation is: \[\text{Concentration} \times \text{Volume (in mm}^3)\]

In our scenario, the calculation would be:\[12,000 \text{ cells/mm}^3 \times 5,000,000 \text{ mm}^3 = 60,000,000,000 \text{ cells}\]This huge number is then often expressed in scientific notation for simplicity, resulting in:\[6.0 \times 10^{10} \text{ cells}\]This number provides an easy-to-read estimate of the total white blood cells in 5 liters of blood, making it clear how concentrations multiply across volume to give absolute numbers.