Imagine a spherical cell with a radius of 10 μm. What is the cell's surface area in μm2? Its volume, in μm3? (Note: For a sphere of radius r, surface area = 4πr2 and volume = 4/3πr3. Remember that the value of π is 3.14.) What is the ratio of surface area to volume for this cell? Now do the same calculations for a second cell, this one with a radius of 20 μm. Compare the surface-to-volume ratios of the two cells. How is this comparison significant to the functioning of cells?

Verified step by step guidance

1. Step 1: Start by calculating the surface area of the first cell (radius = 10 μm). Use the formula for the surface area of a sphere: <math xmlns='http://www.w3.org/1998/Math/MathML'> <mi>A</mi> <mo>=</mo> <mn>4</mn> <mi>&#x3C0;</mi> <msup><mi>r</mi><mn>2</mn></msup> </math>. Substitute r = 10 μm into the formula.
2. Step 2: Next, calculate the volume of the first cell using the formula for the volume of a sphere: <math xmlns='http://www.w3.org/1998/Math/MathML'> <mi>V</mi> <mo>=</mo> <mfrac><mn>4</mn><mn>3</mn></mfrac> <mi>&#x3C0;</mi> <msup><mi>r</mi><mn>3</mn></msup> </math>. Substitute r = 10 μm into the formula.
3. Step 3: Determine the surface area-to-volume ratio for the first cell by dividing the surface area by the volume: <math xmlns='http://www.w3.org/1998/Math/MathML'> <mfrac><mi>A</mi><mi>V</mi></mfrac> </math>. Use the values calculated in Steps 1 and 2.
4. Step 4: Repeat Steps 1 through 3 for the second cell, which has a radius of 20 μm. Use the same formulas for surface area and volume, substituting r = 20 μm.
5. Step 5: Compare the surface area-to-volume ratios of the two cells. Discuss the significance of the comparison in terms of cell function, such as nutrient exchange and waste removal, emphasizing how smaller cells have a higher surface area-to-volume ratio, which is advantageous for efficient transport processes.

Key Concepts

Surface Area and Volume Formulas

The surface area and volume of a sphere are calculated using specific mathematical formulas: surface area = 4πr² and volume = 4/3πr³. These formulas are essential for determining the physical characteristics of spherical cells, which are common in biological systems. Understanding these calculations allows for insights into how cells interact with their environment.

Surface Area-to-Volume Ratio

The surface area-to-volume ratio (SA:V) is a critical concept in biology that describes how the surface area of a cell relates to its volume. A higher SA:V ratio indicates that a cell has more surface area relative to its volume, which facilitates efficient nutrient uptake and waste removal. This ratio is crucial for understanding cellular efficiency and limits on cell size.

Biological Implications of Cell Size

Cell size has significant biological implications, particularly regarding metabolic rates and the efficiency of transport processes. Smaller cells tend to have higher SA:V ratios, allowing for more effective exchange of materials with their environment. This concept is vital for understanding how cell size influences growth, function, and overall organismal physiology.