Consider three spheres with radii of 1 cm, 5 cm, and 10 cm.
Calculate the surface area and the volume of each sphere, and plot the results on a graph with radius on the x-axis and surface area and volume on the y-axis. (Surface area of a sphere = 4πr2; volume of a sphere = (4/3)πr3.)
Explain how the graph shows the relationship between size and surface area to volume ratio.

Verified step by step guidance

1. Step 1: Write down the formulas for the surface area and volume of a sphere. The surface area (A) is given by $A\; =\; 4\pi r2$, and the volume (V) is given by $V\; =\; (4/3)\pi r3$, where r is the radius of the sphere.
2. Step 2: Substitute the given radii (1 cm, 5 cm, and 10 cm) into the formulas for surface area and volume. For each radius, calculate the surface area using $A\; =\; 4\pi r2$ and the volume using $V\; =\; (4/3)\pi r3$. Write down the expressions for each calculation.
3. Step 3: Create a table to organize the results. The table should have three columns: radius (r), surface area (A), and volume (V). Fill in the table with the calculated values for each radius.
4. Step 4: Plot the results on a graph. Use the radius (r) as the x-axis and plot two separate curves on the y-axis: one for surface area (A) and one for volume (V). Ensure the graph is labeled appropriately, with a legend to distinguish between the two curves.
5. Step 5: Analyze the graph to explain the relationship between size and the surface area-to-volume ratio. Note that as the radius increases, the surface area and volume both increase, but the volume grows faster than the surface area. This is because volume is proportional to the cube of the radius ($r3$), while surface area is proportional to the square of the radius ($r2$). Discuss how this affects biological systems, such as the efficiency of nutrient exchange in cells.

Key Concepts

Surface Area and Volume Formulas

The surface area and volume of a sphere are calculated using specific mathematical formulas: the surface area is given by 4πr², and the volume is calculated as (4/3)πr³. These formulas highlight how both surface area and volume increase with the radius of the sphere, but at different rates, which is crucial for understanding their relationship.

Surface Area to Volume Ratio

The surface area to volume ratio (SA:V) is a critical concept in biology and physics, representing how much surface area is available for exchange relative to the volume of an object. As the size of a sphere increases, its volume grows faster than its surface area, leading to a decrease in the SA:V ratio, which can impact processes like diffusion and heat exchange in biological systems.

Graphical Representation of Data

Plotting the surface area and volume against the radius on a graph allows for visual analysis of the relationship between these variables. The graph typically shows that while both surface area and volume increase with radius, the volume increases at a faster rate, illustrating the diminishing surface area to volume ratio as the size of the sphere increases.