Question

The volume of a red blood cell is about \(90.0 \times 10^{-12} \mathrm{cm}^{3} .\) Assuming that red blood cells are spherical, what is the diameter of a red blood cell in millimeters?

Short answer

The diameter of a red blood cell is approximately 0.0554 mm.

Step-by-step solution

Re-arrange the formula for sphere volume to solve for r

Firstly, we need to make 'r' the subject of the formula, which gives us \(r = (\frac{3v}{4\pi})^{\frac{1}{3}}\).

Substitute the volume of the red blood cell into the formula

Given that the volume of the red blood cell is \(90.0 \times 10^{-12} cm^{3}\), replacing 'v' with this value in the formula gives \(r = (\frac{3(90.0 \times 10^{-12})}{4\pi})^{\frac{1}{3}}\).

Calculate the radius r

Evaluating the expression for 'r' gives us a result in centimeters (approximately 0.00277 cm).

Convert radius to diameter in millimeters

Since diameter is twice the radius, the diameter in centimeters is approximately 0.00554 cm. Conversion of this to millimeters (by multiplying with 10 because 1 cm = 10 mm) gives an approximate diameter of 0.0554 mm.

Key concepts

Red Blood Cell Volume

Understanding the volume of a red blood cell is crucial when considering its ability to carry oxygen throughout the body. A typical red blood cell has a volume of approximately 90.0 x 10^{-12} cubic centimeters. This usually small number reflects the minuscule size of these cells that efficiently transport oxygen from the lungs to the rest of the body and help in the return of carbon dioxide. By imagining red blood cells as tiny spheres, like marbles, we can use this approximation to apply mathematical formulas concerning spheres. This simplification aids in calculating other dimensions of the cell, such as its diameter, which is essential for many biological and health science studies.

Geometric Formulas

To find the diameter of a red blood cell, we apply geometric formulas to our sphere. Since the cell's shape is spherical, we use the formula for the volume of a sphere, which is \( V = \frac{4}{3} \pi r^3 \), where \( V \) is the volume and \( r \) is the radius. We need to find the radius from the cell's volume to proceed with solving for the diameter. Rearranging the sphere formula, we obtain \( r = \left( \frac{3V}{4\pi} \right)^{\frac{1}{3}} \). Substituting the given volume into this formula allows us to calculate the radius and subsequently the diameter, by the simple fact that the diameter \( d \) is twice the radius \( r \). This mathematical approach provides a precise measurement based on geometric principles.

Unit Conversion

Often in scientific calculations, we need to convert between different units. In this case, the challenge is to determine the diameter of the red blood cell in millimeters rather than the initial cubic centimeters provided for the volume. After calculating the radius of the red blood cell in centimeters using the earlier formula, we get a value of approximately 0.00277 cm for the radius. To find the diameter, we double this value to arrive at 0.00554 cm. Since 1 centimeter equals 10 millimeters, converting the diameter to millimeters involves simply multiplying by 10. Thus, the diameter of the cell is approximately 0.0554 mm. Unit conversion is a critical step in ensuring that the measurements are in the most practical units for analysis and application.