Question

Consider three spheres with radii of \(1 \mathrm{cm}, 5 \mathrm{cm},\) and \(10 \mathrm{cm} .\) 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 \(=4 \pi r^{2}\); volume of a sphere \(=(4 / 3) \pi r^{3}\).) Explain how the graph shows the relationship between size and surface area/volume ratio.

Short answer

The surface area and volume of the three spheres are calculated as follows: Sphere 1 has a surface area of approximately \(12.57 \mathrm{cm^2}\) and a volume of approximately \(4.19 \mathrm{cm^3}\); Sphere 2 has a surface area of approximately \(314.16 \mathrm{cm^2}\) and a volume of approximately \(523.60 \mathrm{cm^3}\); Sphere 3 has a surface area of approximately \(1256.64 \mathrm{cm^2}\) and a volume of approximately \(4188.79 \mathrm{cm^3}\). When plotted on a graph, both surface area and volume increase with the radius. However, the volume increases at a much faster rate, indicating a decreasing surface area-to-volume ratio as the sphere's size increases. This can have implications in fields such as biology where the surface area-to-volume ratio can affect the efficiency of cellular processes.

Step-by-step solution

Calculate surface areas and volumes

For each sphere, we will calculate its surface area and volume using the given formulas: 1. Surface area: \(A = 4 \pi r^{2}\) 2. Volume: \(V = \frac{4}{3} \pi r^{3}\)

Calculate values for each sphere

Sphere 1: - Radius: 1 cm - Surface area: \(A = 4 \pi (1)^{2} = 4 \pi \approx 12.57 \mathrm{cm^2}\) - Volume: \(V = \frac{4}{3} \pi (1)^{3} = \frac{4}{3} \pi \approx 4.19 \mathrm{cm^3}\) Sphere 2: - Radius: 5 cm - Surface area: \(A = 4 \pi (5)^{2} = 100 \pi \approx 314.16 \mathrm{cm^2}\) - Volume: \(V = \frac{4}{3} \pi (5)^{3} = \frac{4}{3} (125) \pi \approx 523.60 \mathrm{cm^3}\) Sphere 3: - Radius: 10 cm - Surface area: \(A = 4 \pi (10)^{2} = 400 \pi \approx 1256.64 \mathrm{cm^2}\) - Volume: \(V = \frac{4}{3} \pi (10)^{3} = \frac{4}{3} (1000) \pi \approx 4188.79 \mathrm{cm^3}\)

Plot the results on a graph

We will now plot the results on a graph with radius on the x-axis and surface area and volume on the y-axis. Be sure to label the x-axis with the respective radii (1 cm, 5 cm, and 10 cm) and the y-axis with the values for surface area and volume. To make the graph more readable, you can use one color and marker for surface area and another for volume.

Analyze the relationship between size and surface area/volume ratio

From the plotted graph, we can observe that as the radius of the sphere increases, both the surface area and the volume increase. However, the volume increases at a much faster rate compared to the surface area. This can be seen through the steeper curve plotted for the volume. The volume of a sphere has a cubic relationship with the radius, while the surface area has a quadratic relationship. As the sphere's size (or radius) increases, the ratio of its surface area to its volume decreases. This is due to the fact that the volume is increasing at a faster rate than the surface area. This can have important implications in various fields, such as biology, where the surface area-to-volume ratio can affect the efficiency of cellular processes.

Key concepts

Sphere Calculations

Understanding the properties of spheres can be fascinating, especially when you start calculating aspects like surface area and volume. Spheres are three-dimensional objects, where every point on the surface is equidistant from the center.

Here's a quick guide on calculating these properties for any given sphere:

• Surface Area: The formula to calculate the surface area of a sphere is \( A = 4 \pi r^{2} \). This equation represents the total area that the surface of the sphere covers.
• Volume: For the volume, the strategy involves filling the sphere, calculated with \( V = \frac{4}{3} \pi r^{3} \). This formula gives you the total space occupied by the sphere inside.

In the exercise, we calculated the surface area and volume for three spheres with radii of 1 cm, 5 cm, and 10 cm. By substituting the radius into the formula, you can determine each sphere's dimensions. For example, if a sphere has a radius of 5 cm, inserting this into the surface area formula gives \( A = 4 \pi (5)^{2} \approx 314.16 \mathrm{cm^2} \), while the volume becomes \( V = \frac{4}{3} \pi (5)^{3} \approx 523.60 \mathrm{cm^3} \).

These calculations are crucial for understanding the relationships and differences in size across different spheres.

Surface Area to Volume Ratio

One intriguing aspect of spherical calculations is how the surface area compares to the volume, a relationship crucial in many natural and technological processes. As spheres grow larger, this ratio changes dramatically, affecting their appearances and interactions.

The ratio of surface area to volume is computed by dividing the surface area by the volume \( \frac{A}{V} \). As the radius increases, this ratio decreases, meaning the volume grows faster than the surface area.
For small objects, like biological cells, a high surface area-to-volume ratio is beneficial because it allows efficient exchange of materials, necessary for rapid cellular processes. However, larger organisms or structures may prefer a lower ratio to better retain heat or moisture.
Visualizing this relationship on a graph reveals a steep curve for volume as compared to surface area, suggesting that doubling the radius results in a greater volumetric increase compared to surface area. This provides insights into why large spheres, such as planets or fruits, often have less surface structure relative to their volume.

Mathematical Modeling

Mathematical modeling using spheres is a powerful tool for understanding real-world phenomena. By using spheres to model, you can gain insights into systems ranging from atoms to planets.

In practical terms, mathematical modeling with spheres allows:

• Prediction: Figuring out how characteristics like size or density change can help predict behavior in larger systems, such as weather modeling or astrophysics.
• Optimization: Understanding sphere dynamics allows for better design of things like containers, tanks, or even living organisms, maximizing volume while minimizing surface area.
• Analysis: Using the relationships identified, such as surface area to volume ratios, you can analyze and simulate scenarios efficiently, useful in fields like material science or engineering design.

By plotting spheres' calculated surface areas and volumes against their radii, you can identify trends and dependencies that translate to real-world applications. In computational simulations, this fundamental understanding helps build more accurate and efficient models, aiding in advancements across a range of sciences. Utilizing such mathematical principles makes complex systems more approachable and comprehensible.