Problem 57 - Biocalculus Calculus for the Life Sciences

Question

An oil storage tank ruptures at time \(t=0\) and oil leaks from the tank at a rate of \(r(t)=100 e^{-0.01 t}\) liters per minute. How much oil leaks out during the first hour?

Step-by-Step Solution

Answer

About 4512 liters of oil leaks out during the first hour.

Step 1: Understand the Problem

We need to find the total amount of oil that leaks out of the tank during the first hour (from \(t=0\) to \(t=60\) minutes). The rate of leakage is given by the function \(r(t) = 100 e^{-0.01t}\) liters per minute.

Step 2: Set Up the Integral

The total amount of oil that leaks out over the time interval \([0, 60]\) is given by the integral of the rate function \(r(t)\) from 0 to 60. This integral is \( \int_0^{60} 100 e^{-0.01t} \, dt \).

Step 3: Integrate the Function

Perform the integration \( \int 100 e^{-0.01t} \, dt \). The integral of \(e^{-0.01t}\) is \(-100 e^{-0.01t}/0.01 + C = -10000 e^{-0.01t} + C\). We do not need the constant \(C\) since we are computing a definite integral.

Step 4: Evaluate the Definite Integral

Evaluate \(-10000 e^{-0.01t}\) from \(t = 0\) to \(t = 60\): - Compute \(-10000 e^{-0.01 \times 60}\) and \(-10000 e^{-0.01 \times 0}\). - So, \(-10000 e^{-0.6}\) and \(-10000 e^{0}\) respectively. - The result is \(-10000(e^{-0.6} - 1)\).

Step 5: Calculate the Result

Evaluate \(-10000(e^{-0.6} - 1)\): - Use a calculator to find \(e^{-0.6} \approx 0.5488\). - Substitute it into the expression: \(-10000(0.5488 - 1) \approx -10000(-0.4512) \approx 4512\). - Therefore, the total amount of oil that leaks out during the first hour is approximately 4512 liters.

Key Concepts

Definite Integral

In calculus, the concept of a definite integral is a powerful tool used to calculate the total accumulation of a quantity over a specific interval. When given a rate of change or a function describing how a quantity changes over time, you can use a definite integral to find the total amount that has changed during a period. In our example, we are dealing with oil leakage over time. Here, the rate of oil leakage is provided by a function that varies as time progresses.

The definite integral is represented mathematically by \ \[\int\] \ with lower and upper limits, which define the interval of interest. For example, calculating the total oil leaked out from a ruptured tank over the first hour is done by setting the interval from zero to sixty minutes.

To perform a definite integral, follow these steps:

• Set up the integral with the limits of the interval (in this case, from 0 to 60 minutes) and the rate function.
• Perform the integration using calculus rules to find an antiderivative.
• Evaluate this antiderivative at the upper and lower limits, and find the difference.

By doing this, you can find how much of a certain quantity—say, oil—has accumulated over the given time frame.

Exponential Functions

Exponential functions are a type of mathematical function in which the variable appears in the exponent. These functions are essential in modeling scenarios where quantities grow or decay at constant proportional rates. In our exercise, the oil leakage rate is modeled using an exponential function: \ \[r(t) = 100 e^{-0.01t}\].

Exponential decay occurs when a quantity decreases at a rate proportional to its current value, which is the case with our leakage rate. The decay parameter in the exponent, \ \[-0.01\], determines the rapidity of this decay. Smaller negative numbers result in slower decay rates.

Understanding exponential functions helps in examining:

• Initial rates and how quickly they change over time.
• The behavior of quantities under proportional rate changes.
• Predicting future values of the quantity, given the function.

In practical applications such as this one, exponential functions are used to simulate phenomena like cooling, population dynamics, and radioactive decay, making them an essential part of understanding real-world processes.

Calculus Application

Calculus provides tools to solve real-world problems involving dynamic systems, where quantities change continuously over time. In our example, calculus allows us to precisely determine the amount of oil leaked from the tank by integrating the rate function over time. This integration is essentially the summation of infinitesimally small amounts over a continuous interval, producing an accurate total.

Applications of calculus like this are seen across numerous fields:

• Engineering: Calculating load distribution, optimizing design parameters.
• Physics: Modeling the motion of bodies under various forces over time.
• Economics: Measuring consumer surplus, growth, and other economic indicators.
• Biology: Predicting changes in populations over time and understanding biological processes.

With calculus, particularly using integrals, we can solve complex problems involving changing rates, such as fluid dynamics, chemical reactions, or even financial predictions. The mathematical clarity and insight provided by calculus make it indispensable for rigorous analysis and precision in these applications.