Question

How large, in cubic centimeters, is the volume of a red blood cell (in \(\mathrm{cm}^{3}\) ) if the cell has a cylindrical shape with a diameter of \(6 \times 10^{-6} \mathrm{~m}\) and a height of \(2 \times 10^{-6} \mathrm{~m}\) ? What is the volume in pL?

Short answer

The volume of the red blood cell is approximately 56.55 picoliters (pL).

Step-by-step solution

Convert units

The diameter and height of the red blood cell are given in meters, but we need the volume in cubic centimeters. First, let's convert these measurements to centimeters. Convert the diameter:\[6 \times 10^{-6} \text{ m} = 6 \times 10^{-6} \text{ m} \times 100 \text{ cm/m} = 6 \times 10^{-4} \text{ cm}\]Convert the height:\[2 \times 10^{-6} \text{ m} = 2 \times 10^{-6} \text{ m} \times 100 \text{ cm/m} = 2 \times 10^{-4} \text{ cm}\]

Calculate the radius

The radius is half the diameter. To find the radius in centimeters:\[ \text{Radius} = \frac{\text{Diameter}}{2} = \frac{6 \times 10^{-4} \text{ cm}}{2} = 3 \times 10^{-4} \text{ cm} \]

Apply the volume formula for a cylinder

The volume \( V \) of a cylinder is given by the formula:\[ V = \pi r^2 h \] where \( r \) is the radius and \( h \) is the height. Plugging in the values:\[ V = \pi \times (3 \times 10^{-4} \text{ cm})^2 \times 2 \times 10^{-4} \text{ cm} \]

Perform the calculations

Calculate \( (3 \times 10^{-4})^2 \):\[ (3 \times 10^{-4})^2 = 9 \times 10^{-8} \]Now plug in the values to get:\[ V = \pi \times 9 \times 10^{-8} \text{ cm}^2 \times 2 \times 10^{-4} \text{ cm} \]\[ V = 18\pi \times 10^{-12} \text{ cm}^3 \]Using \( \pi \approx 3.14159 \), we get:\[ V \approx 18 \times 3.14159 \times 10^{-12} \text{ cm}^3 \approx 56.55 \times 10^{-12} \text{ cm}^3 \]

Convert volume to picoliters (pL)

1 cubic centimeter equals 1 milliliter (mL) and 1 milliliter equals 1,000,000 picoliters (pL). Therefore, to convert the volume from cubic centimeters to picoliters:\[ 56.55 \times 10^{-12} \text{ cm}^3 \times 1,000,000 \text{ pL/mL} = 56.55 \text{ pL} \]

Key concepts

Cubic Centimeters

A cubic centimeter (cm³) is a unit of volume that is commonly used in various scientific measurements. Imagine a cube where each side measures 1 centimeter; the amount of space inside this cube is 1 cubic centimeter. This unit is often used in biology, chemistry, and medicine. For example, when measuring the volume of a cell or liquid sample in a lab.

In many cases, volume in cubic centimeters can also be linked to other units. A single cubic centimeter is equivalent to 1 milliliter (mL). This equivalence aids in making quick and easy conversions between these two units, especially in scientific settings where mL is extensively used.

Unit Conversion

Unit conversion is essential in solving problems where measurements are given in various units, yet a unified unit is needed. This often requires multiplying by conversion factors to change units appropriately.

For example, converting meters to centimeters involves converting bigger units to smaller ones. There are 100 centimeters in a meter, so to convert meters to centimeters, simply multiply the number of meters by 100:

• For the diameter: \[ 6 \times 10^{-6} \text{ m} \times 100 = 6 \times 10^{-4} \text{ cm} \]
• For the height: \[ 2 \times 10^{-6} \text{ m} \times 100 = 2 \times 10^{-4} \text{ cm} \]

This conversion ensures accurate and consistent calculations when applying formulas involving measurements, like calculating volume in cubic centimeters.

Cylinder Volume

The volume of a cylinder-shaped object can be calculated using the formula:\[ V = \pi r^2 h \]where \( V \) is the volume, \( r \) is the radius of the cylinder's base, and \( h \) is its height. This formula works because the base of a cylinder is a circle, and the volume is essentially the area of this circular base multiplied by the cylinder's height.

In our case, the red blood cell is cylindrical, so we apply this formula. After converting to centimeters:

• The radius is half of the diameter: \[ \text{Radius} = \frac{6 \times 10^{-4} \text{ cm}}{2} = 3 \times 10^{-4} \text{ cm} \]
• The height is \( 2 \times 10^{-4} \text{ cm} \)
• Substituting into the formula gives:\[ V = \pi \times (3 \times 10^{-4} \text{ cm})^2 \times 2 \times 10^{-4} \text{ cm} \]

This calculation helps to find the volume in cubic centimeters.

Mathematical Calculations

Completing mathematical calculations involves steps that require attention to detail, especially when dealing with very small or very large numbers. For a red blood cell, this entails working with powers of ten and the constant \( \pi \).

First, calculate the square of the radius in scientific notation:

• \[ (3 \times 10^{-4})^2 = 9 \times 10^{-8} \]

Next, substitute all known values into the cylinder volume formula:

• \[ V = \pi \times 9 \times 10^{-8} \times 2 \times 10^{-4} \]
• \[ V = 18\pi \times 10^{-12} \text{ cm}^3 \]
• Approximating \( \pi \) as 3.14159 results in:\[ V \approx 56.55 \times 10^{-12} \text{ cm}^3 \]

This volume can then be converted to picoliters by multiplying by 1,000,000, leading to a value of 56.55 pL.