Problem 5 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

At 1: 00 a.m. an oil pipe line bursts and starts releasing oil into a lake at the rate of 2 cubic meters per hour. At 2: 00 a.m., a second oil pipe line bursts and also starts releasing oil into the lake at the rate of 3 cubic meters per hour. a. How much oil is in the lake at 1: 00,1: 30,2: 00,2: 30,3: 00,\(3: 30,\) and \(4: 00 ?\) b. Let \(\mathrm{T}(\mathrm{x})\) be the total amount of oil in the lake at time \(x\). Draw a graph of \(T\). c. Write equations describing the total amount of oil, \(T(x),\) in the lake for each time \(x\) between 1:00 a.m. and 4: 00 a.m. d. Compute \(T^{\prime}\).

Step-by-Step Solution

Answer

a) Oil at those times: 0, 1, 2, 4.5, 7, 9.5, 12 cubic meters. b) Graph is piecewise linear: slope 2, then 5. c) Equations: before 2 a.m., \(T(x) = 2(x-1)\); after, \(T(x) = 5x - 4\). d) \(T'(x) = 2\) before 2 a.m., \(T'(x) = 5\) after.

Step 1: Calculate oil at different times

- **At 1:00 a.m.**: The first pipeline begins releasing oil, but no oil has accumulated yet.- **At 1:30 a.m.**: The first pipeline releases oil for 0.5 hours at 2 cubic meters per hour: \(2 \times 0.5 = 1\) cubic meter.- **At 2:00 a.m.**: The first pipeline releases oil for 1 hour: \(2 \times 1 = 2\) cubic meters.- **At 2:30 a.m.**: The first pipeline has been releasing for 1.5 hours: \(2 \times 1.5 = 3\) cubic meters. The second pipeline starts at 2:00 a.m. and releases for 0.5 hours: \(3 \times 0.5 = 1.5\) cubic meters. Total is \(3 + 1.5 = 4.5\) cubic meters.- **At 3:00 a.m.**: The first pipeline has released for 2 hours: \(2 \times 2 = 4\) cubic meters. The second pipeline has released for 1 hour: \(3 \times 1 = 3\) cubic meters. Total is \(4 + 3 = 7\) cubic meters.- **At 3:30 a.m.**: The first pipeline has released for 2.5 hours: \(2 \times 2.5 = 5\) cubic meters. The second pipeline has released for 1.5 hours: \(3 \times 1.5 = 4.5\) cubic meters. Total is \(5 + 4.5 = 9.5\) cubic meters.- **At 4:00 a.m.**: The first pipeline has released for 3 hours: \(2 \times 3 = 6\) cubic meters. The second pipeline has released for 2 hours: \(3 \times 2 = 6\) cubic meters. Total is \(6 + 6 = 12\) cubic meters.

Step 2: Graph the total amount of oil, T(x)

The graph of \(T(x)\) is a piecewise function:- **From 1:00 a.m. to 2:00 a.m.**: The function is linear with slope 2, starting from 0. Graph this as a straight line starting at point (1,0) and ending at point (2,2).- **From 2:00 a.m. to 4:00 a.m.**: The function's slope changes to 5 (combination of both pipelines). Graph this as a straight line with a steeper slope starting from point (2,2) to point (4,12). This shows that after 2:00 a.m., the oil amount increases more rapidly.

Step 3: Write equations for T(x)

According to each time interval:- **For 1:00 a.m. to 2:00 a.m.**: The equation is \(T(x) = 2(x - 1)\) cubic meters, starting at 0 cubic meters at 1:00 a.m.- **For 2:00 a.m. to 4:00 a.m.**: The equation becomes \(T(x) = 2(x - 1) + 3(x - 2)\) cubic meters which simplifies to \(T(x) = 5x - 4\). At 2:00 a.m., you can see this continuity, where it starts at 2 cubic meters and grows with a rate of 5 cubic meters per hour.

Step 4: Compute T'

The derivative \(T'(x)\) represents the rate of oil entering the lake and is piecewise:- **For 1:00 a.m. to 2:00 a.m.**: \(T'(x) = 2\) cubic meters per hour, as only the first pipeline feeds oil.- **For 2:00 a.m. to 4:00 a.m.**: \(T'(x) = 2 + 3 = 5\) cubic meters per hour, with both pipelines adding oil after 2:00 a.m.

Key Concepts

Understanding Piecewise Functions

Piecewise functions are mathematical expressions with different defined sub-functions within particular intervals. In problems like the oil spill modeling, piecewise functions help describe situations where the behavior changes at specific times or conditions.
Take the oil spill scenario: Two pipelines begin to leak oil into a lake at different times. To model the total amount of oil in the lake, we use a piecewise function.

• Between 1:00 a.m. and 2:00 a.m., only the first pipeline is active. The rate of oil entering the lake is constant, represented by the linear function with a slope of 2.
• After 2:00 a.m., the second pipeline starts leaking oil. The rate becomes a combined effect of both pipelines, requiring a different function to accurately depict this increase in oil.

These functions are concatenated and allow us to see the changes in the total amount of oil over time. By understanding the segments separately, and then combining them, one can model complex events effectively and realistically.

Exploring the Rate of Change

The rate of change is a crucial concept in calculus and is fundamental to understanding the dynamics of oil spill modeling. It provides insights into how quickly or slowly something is changing over time.
In the context of the oil leak, the rate of change tells us how the total volume of oil in the lake is increasing at any given moment.

• For the first part of the event, from 1:00 a.m. to 2:00 a.m., the rate of change is constant at 2 cubic meters per hour. This is because only one pipeline is leaking oil. The derivative, which measures this rate, remains uniform during this interval.
• From 2:00 a.m. onwards, the rate of change rises to 5 cubic meters per hour. This increased rate describes the combined output of two pipelines leaking simultaneously. Again, the derivative reflects this revised rate, showing a different behavior in the system.

Understanding the rate helps to anticipate the conditions in scenarios such as oil spills, informing strategies for containment and cleanup.

Oil Spill Modeling for Environmental Impact Assessment

Oil spill modeling involves predicting the spread and impact of spilled oil, vital for planning effective response measures. Through calculus, such models can estimate the spill’s progression over time, helping to forecast wider environmental consequences.
In the given task, oil spill modeling begins with calculating how much oil enters the lake over distinct time intervals.

• Initially, the model uses straightforward calculations based on the rate the oil enters the lake. This involves accumulation at set intervals (e.g., every 30 minutes), as seen in the problem's initial calculations.
• Subsequently, by employing the derived piecewise functions, the model captures a comprehensive view of oil totals over the stated time frame. Graphs of these functions display the timeline of the oil spill, showcasing both linear relationships and shifts in accumulation rates.

This modeling approach allows environmental scientists and response teams to visualize potential scenarios, adapt quickly to changes, and minimize the environmental impacts of oil spills effectively.