Problem 15 - Applied Calculus

Question

The fallout, \(V\) (in kilograms per square 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. Over time, fallout is initially larger but decreases similarly as it spreads.

Step 1: Understand the Function

The function model given is \( V = f(d,t) = (\sqrt{t}) e^{-d} \), where \( V \) is in kilograms per square kilometer and depends on distance \( d \) and time \( t \). Here, \( e^{-d} \) represents an exponential decay with respect to distance \( d \), and \( \sqrt{t} \) adjusts the scale based on time \( t \).

Step 2: Calculate for Cross-Section at t=1

Substitute \( t = 1 \) into the function: \( V_1 = (\sqrt{1}) e^{-d} = e^{-d} \). This function decreases exponentially as distance \( d \) increases from the volcano, indicating that fallout decreases rapidly with distance.

Step 3: Calculate for Cross-Section at t=2

Substitute \( t = 2 \) into the function: \( V_2 = (\sqrt{2}) e^{-d} \). Here, the scale is multiplied by \( \sqrt{2} \), suggesting that at \( t = 2 \), the initial fallout amount is higher compared to \( t = 1 \), but it still decreases exponentially with distance \( d \).

Step 4: Sketch the Graphs

On the same axes, plot both cross-sections: \( V_1 = e^{-d} \) and \( V_2 = \sqrt{2} e^{-d} \). Both graphs should start at different initial values at \( d = 0 \) (\( 1 \) for \( V_1 \) and \( \sqrt{2} \) for \( V_2 \)) and drop off quickly as \( d \) increases.

Step 5: Analyze and Compare the Graphs

The graphs show that fallout decreases exponentially as distance from the volcano increases. For \( t = 2 \), the fallout starts off larger than at \( t = 1 \), due to the factor of \( \sqrt{2} \), but both decrease at the same rate with distance. Over time, the fallout further from the volcano is less as it settles and spreads out.

Key Concepts

Volcanic Fallout

Volcanic fallout refers to the ash and debris that are ejected into the air during a volcanic explosion and eventually settle back to the Earth's surface. This can include small particles like ash, as well as larger volcanic rocks. Fallout from a volcanic explosion can have significant environmental and health impacts, and understanding how it spreads is crucial.
The given model, \(V = f(d, t) = (\sqrt{t}) e^{-d}\), helps to predict the amount of fallout in kilograms per square kilometer. The key factors are:

• Distance \(d\) from the volcano.
• Time \(t\) since the explosion.

For example, as shown in the exercise, when \(t = 1\) and \(t = 2\), the volcanic fallout decreases exponentially with distance. This means the closer you are to the volcano, the more fallout you experience shortly after the eruption. Over time, the fall accumulates at a higher rate initially but then tapers off as it spreads out further from the source, hence affecting areas nearer to the volcano more severely.

Distance-Decay Relationship

The distance-decay relationship is a fundamental concept in many fields, demonstrating how certain effects lessen over distance. For volcanic fallout, this relationship means that as you move further from the volcano, the intensity of fallout decreases. This is modeled in the function \(V = (\sqrt{t}) e^{-d}\), where the term \(e^{-d}\) introduces exponential decay.
In practical terms:

• As the distance \(d\) grows, the value of \(V\) falls sharply. The function \(e^{-d}\) ensures that even a small increase in distance can significantly reduce the impact of fallout.
• This principle helps explain why cities and habitats located closer to a volcano are at a higher risk during an explosion compared to those situated further away.

This distance-decay phenomenon is crucial in planning evacuation and emergency responses during and after volcanic eruptions.

Graph Sketching

Graph sketching is a technique used to visualize mathematical relationships, like how volcanic fallout changes over time and distance. In this exercise, you were asked to sketch the graphs of the function \(f(d, t) = (\sqrt{t}) e^{-d}\) for two different scenarios: at \(t = 1\) and \(t = 2\).
Here's what the graphs tell us:

• For \(t = 1\), the function simplifies to \(V_1 = e^{-d}\). This plot starts at a value of 1 when \(d = 0\) and decreases quickly as distance increases.
• For \(t = 2\), the function becomes \(V_2 = \sqrt{2} e^{-d}\). Here, the starting value at \(d = 0\) is higher, at \(\sqrt{2}\), but the rate of decrease with distance remains the same.

These cross-sections show how fallout varies but decreases consistently over distance. Sketching these helps visualize the rapid decay, reinforcing that the impact is much heavier near the volcanic source. Sketches give a graphical representation of theoretical predictions, aiding both understanding and communication of these complex phenomena.