Problem 15 - Applied Calculus

Question

The fallout, \(V\) (in kilograms per square a kilometer), from a volcanic explosion depends on the distance, \(d\), from the volcano and the time, \(t\), since the explosion: $$V=f(d, t)=(\sqrt{t}) e^{-d}$$ On the same axes, graph cross-sections of \(f\) with \(t=1\), and \(t=2\). As distance from the volcano increases, how does the fallout change? Look at the relationship between the graphs: how does the fallout change as time passes? Explain your answers in terms of volcanoes.

Step-by-Step Solution

Answer

As distance increases, fallout decreases exponentially. With time, fallout increases due to the dispersion.

Step 1: Analyze the function

The function given is \( V = f(d, t) = (\sqrt{t}) e^{-d} \). It represents fallout \( V \) as a function of distance \( d \) and time \( t \). The exponential term \( e^{-d} \) suggests that the fallout decreases exponentially with an increase in distance \( d \). The \( \sqrt{t} \) term indicates that fallout increases with time because the square root is a slowly increasing function.

Step 2: Set up cross-sections for graphing

We are asked to graph cross-sections of \( f \) for \( t = 1 \) and \( t = 2 \). This means substituting these values of \( t \) into the function and plotting \( V \) as a function of \( d \). For \( t=1 \), \( V = f(d, 1) = (\sqrt{1}) e^{-d} = e^{-d} \). For \( t=2 \), \( V = f(d, 2) = (\sqrt{2}) e^{-d} \).

Step 3: Plot the graphs

Draw two plots of \( V \) versus \( d \) where \( V = e^{-d} \) for \( t=1 \) and \( V = \sqrt{2} e^{-d} \) for \( t=2 \). Both plots will start at their respective maximum values \( V = \sqrt{t} \) when \( d = 0 \), and both will decrease as \( d \) increases. However, the curve for \( t=2 \) will start higher because \( \sqrt{2} > 1 \).

Step 4: Describe fallout change with distance

From the cross-section graphs, observe that the fallout \( V \) decreases continuously as the distance \( d \) increases due to the exponential decay \( e^{-d} \), indicating that the fallout diminishes significantly the farther away it is from the volcano.

Step 5: Relate graphs to time change

By comparing the two graphs, note that as time increases from \( t=1 \) to \( t=2 \), the initial fallout starts from a higher value \( \sqrt{2}e^{-d} \). Therefore, as time since the explosion increases, the maximum potential fallout increases, representing that the initial distribution of material becomes more dispersed and potentially less concentrated near the volcano.

Key Concepts

Exponential Decay

Volcanic fallout modeling often involves understanding exponential decay, which describes how rapidly the fallout amounts decrease with distance from the volcano. In our function, the term \( e^{-d} \) captures this behavior. Here, \( d \) is the distance from the volcano, and as \( d \) increases, \( e^{-d} \) decreases exponentially. This means the further you move away from the source of the explosion, the less the fallout you will encounter.

• Exponential decay shows a rapid drop at first, which then slows down.
• This reflects the natural spreading and thinning of volcanic ash as it moves away from the explosion center.

Such behavior is typical in natural processes where energy or materials disperse over time and space. Recognizing exponential decay allows us to calculate and predict regional impact zones with more accuracy.

Distance and Time Dependency

The distance and time dependency of volcanic fallout is crucial in understanding how ash spreads and settles. Our function \( V = f(d, t) = (\sqrt{t}) e^{-d} \) indicates that fallout is influenced by both distance and time.

• The term \( \sqrt{t} \) signifies that as time progresses, fallout distribution changes. This dependency ensures that measurements account for how fallout behaves over time.
• At any time \( t \), the fallout decreases exponentially with distance \( d \), illustrating how distance attenuates the fallout effect.

Understanding these dependencies helps us predict how ash becomes more dispersed over time, offering vital insights into how to mitigate the fallout's effects using distance as a buffer.

Cross-Sectional Analysis

In volcanic fallout modeling, cross-sectional analysis involves examining specific `slices` of our function at given time points. By looking at them, we can understand how fallout patterns develop over time. In our exercise, we are tasked with comparing cross-sections at \( t=1 \) and \( t=2 \).

• A cross-section at \( t=1 \): \( V = e^{-d} \), reflects the initial fallout distribution soon after the explosion.
• A cross-section at \( t=2 \): \( V = \sqrt{2} e^{-d} \), indicates increased maximum values, showing increased fallout potential further from the volcano.

Both cross-sections begin at \( d=0 \) with different intensities but express a common pattern. As time progresses, the initial intensity at the origin becomes more pronounced with a higher maximum fallout capability. This analysis aids in visualizing and drawing specific conclusions about expected fallout at various time intervals post-explosion. Hence, cross-sectional analysis in this context can provide pivotal information for disaster preparedness and response planning.