Question

In cosmic Calculations 6.1 , we calculated that the volume of bacteria after 120 doublings would be \(1.3 \times 10^{15} \mathrm{m}^{3} .\) The total surface area of Earth is about \(5.1 \times 10^{14} \mathrm{m}^{2} .\) Use these facts to calculate the average depth of the bacteria at that time, if we spread them evenly over Earth's entire surface.

Short answer

The average depth of the bacteria would be 0.255 meters.

Step-by-step solution

Understand the Given Information

The problem provides that after 120 doublings, the volume of bacteria is \(1.3 \times 10^{15} \text{ m}^3\). The surface area of Earth is \(5.1 \times 10^{14} \text{ m}^2\). We need to find the average depth of bacteria when spread evenly over Earth's surface.

Identify the Formula for Depth

The depth can be calculated using the formula for volume distributed over an area: \(\text{Depth} = \frac{\text{Volume}}{\text{Surface Area}}\). Here, volume refers to the volume of bacteria and surface area to Earth's surface area.

Plug in the Values

Insert the given values into the formula: \[\text{Depth} = \frac{1.3 \times 10^{15} \text{ m}^3}{5.1 \times 10^{14} \text{ m}^2}\].

Perform the Calculation

Divide the volume of the bacteria by the surface area of the Earth: \[\text{Depth} = \frac{1.3}{5.1} \times 10^{15-14} \text{ m}\], which simplifies to: \[\text{Depth} = 0.255 \text{ m}\].

Interpret the Result

The calculated depth suggests that if the bacteria were spread evenly, the average depth covering Earth's surface would be approximately 0.255 meters.

Key concepts

Doubling Time in Biology

Doubling time in biology refers to the period it takes for a population to double in size. This is an essential concept when studying the growth rate of organisms, such as bacteria. In the context of bacteria, this time can vary depending on the species and environmental conditions. For instance, some bacteria might double in as little as 20 minutes under optimal conditions, while others take longer. Understanding doubling time helps scientists predict how quickly a population will grow. It also serves as a critical factor in assessing how fast resources (like nutrients) will be consumed. With each doubling, the number of bacteria increases. If you start with one bacterium and it doubles 120 times, the population will grow exponentially, highlighting the power and speed of bacterial reproduction.

Earth's Surface Area

The Earth's surface area is a large but finite space. It covers approximately \(5.1 \times 10^{14}\) square meters. This area includes all the land and oceans combined, presenting a significant expanse.Utilizing Earth's surface area in calculations allows us to visualize and contextualize large-scale phenomena. For instance, spreading a large volume of objects, such as bacteria, evenly across the Earth's surface offers insights into density and impact. Moreover, this measurement serves as a critical tool in environmental science, helping researchers assess the distribution of resources and biodiversity across the globe.Understanding the enormity of Earth's surface area helps us appreciate the magnitude of covering it entirely with a particular substance or population, such as in the bacterial spread scenario.

Depth Calculation

Calculating depth can involve distributing a volume across a designated area. In our context, when bacteria cover the Earth's surface, calculating how deep that layer would be is essential for understanding density and potential impacts. The formula used here is straightforward:

• \(\text{Depth} = \frac{\text{Volume}}{\text{Surface Area}}\)

This equation helps find out how thick a layer is when a given volume (in this case, bacteria) is spread over a certain surface area. By applying it, we determine how things spread on a large scale. In general, this type of calculation highlights the significance of having accurate measurements of both volume and area to derive meaningful results about a given distribution.

Volume Distribution

Volume distribution explores how a substance is spread across an area. This concept is crucial in understanding scenarios where a material or population needs to cover a space, such as bacteria distributing across Earth. When distributing volume, it is key to know both the total volume and the surface area it covers. In cases like bacteria, understanding how evenly they spread affects predictions about their impact, such as the potential depth of coverage. In biological contexts, volume distribution could apply to not just how organisms spread spatially, but also how nutrients or toxins disperse within ecosystems. Analyzing volume distribution involves thinking about both the environmental implications and the logistical considerations of such spread. It informs decisions and strategies for managing ecosystems or urban spaces.