Question

A spherical cell has a radius of \(34 \mu \mathrm{m}\). What is its surface area/volume ratio? a. 0.088 b. 0.12 c. 11.3 d. 55.7 e. 127

Short answer

The surface area to volume ratio of the sphere is approximately 0.088. Hence, option a. 0.088 is the correct answer.

Step-by-step solution

Compute the Surface Area of the Sphere

The formula for the surface area of a sphere is \(4\pi r^2\), where r represents the radius of the sphere. Substituting the given radius (\(34 \mu m\)), the surface area of the cell is \( 4\pi(34)^2 \approx 14526 \mu m^2\).

Compute the Volume of the Sphere

The formula for the volume of a sphere is \(\frac{4}{3}\pi r^3\), where r is the radius. Substituting the given radius (\(34 \mu m\)), the volume of the cell (~ cell's volume) will be \(\frac{4}{3}\pi(34)^3 \approx 164509 \mu m^3\).

Compute the Ratio

Next is to compute the surface area/volume ratio. The formula for this ratio is \(\frac{Surface Area}{Volume}\). Substituting the surface area (\(14526 \mu m^2\)) and the volume (\(164509 \mu m^3\)), the ratio is \(\frac{14526}{164509} \approx 0.088\).

Key concepts

Spherical Cell

A spherical cell is a cell that has the shape of a sphere, meaning it is perfectly round like a ball. This shape is common in many biological contexts, and it allows for some important mathematical simplifications when calculating properties like surface area and volume. The spherical shape is significant because it provides the maximum volume for a given surface area, making it an efficient shape for cellular functions. This efficiency is crucial for processes such as diffusion and osmosis, which rely heavily on surface area and volume interactions.

Radius

The radius of a spherical cell is a line segment that extends from the center of the cell to any point on its surface. It is a crucial measurement for calculating other important properties of the sphere, such as its surface area and volume. If you know the radius, you can easily determine these properties using mathematical formulas. For instance, in our exercise, the radius of the spherical cell is given as \(34 \mu \text{m}\). Knowing this allows us to calculate both the surface area and the volume effectively. Thus, the radius is foundational to understanding the cell's geometry.

Surface Area of Sphere

The surface area of a sphere, which represents the total area that covers the cell, can be calculated using the formula \(4\pi r^2\). Here, \(r\) is the radius of the sphere. In our example, substituting the radius \(34 \mu \text{m}\) gives us the surface area as \(4\pi(34)^2 \approx 14526 \mu \text{m}^2\). This calculation is essential because the surface area affects how a cell interacts with its environment, including nutrient uptake and waste removal. A larger surface area relative to volume generally indicates more efficient cellular exchange.

Volume of Sphere

The volume of a sphere determines how much space the cell occupies and is calculated using the formula \(\frac{4}{3}\pi r^3\). The radius \(r\) is again the vital component here. With a radius of \(34 \mu \text{m}\), the volume of the cell becomes \(\frac{4}{3}\pi(34)^3 \approx 164509 \mu \text{m}^3\). The volume is critical for understanding how much "content" or cytoplasm the cell can hold. The relationship between surface area and volume impacts metabolic processes, making this calculation crucial for biological understanding.