Question

A cylindrical tank is filled with water to a depth of 9 meters. At \(t=0,\) a drain in the bottom of the tank is opened and water flows out of the tank. The depth of water in the tank (measured from the bottom of the tank) \(t\) seconds after the drain is opened is approximated by \(d(t)=(3-0.015 t)^{2},\) for \(0 \leq t \leq 200 .\) Evaluate and interpret \(\lim _{t \rightarrow 200^{-}} d(t)\).

Short answer

Answer: 0 meters

Step-by-step solution

Identify the function

The function that represents the depth of water in the tank \(t\) seconds after the drain is opened is \(d(t) = (3-0.015t)^2\).

Write the limit to be evaluated

We are asked to find \(\lim_{t \rightarrow 200^{-}} d(t)\).

Substitute the function into the limit

Now, we substitute the function \(d(t)\) into the limit: \(\lim_{t \rightarrow 200^{-}} (3-0.015t)^2\).

Plug in the value

Since the function \(d(t)\) is continuous at \(t=200\), we can substitute the value \(t=200\) directly into the expression: \((3-0.015(200))^2\).

Calculate the result

Now, simplify the expression: \((3-3)^2 = (0)^2 = 0\).

Interpret the result

As a result, \(\lim_{t \rightarrow 200^{-}} d(t) = 0\). This means that as the time approaches 200 seconds from the left, the depth of water in the tank approaches 0 meters. In other words, at about 200 seconds, the tank will be almost empty.

Key concepts

Calculus

Calculus is an advanced field of mathematics that involves the study of change and motion. One of its primary functions is to understand the behavior of functions and how they can be applied to real-world situations. There are two main branches of calculus: differential calculus, which concerns the rate of change and slopes of curves, and integral calculus, which focuses on finding the total size or value, such as areas under curves.

• Differential calculus uses derivatives to determine rates of change.
• Integral calculus uses integrals to calculate areas and volumes.

These concepts have applications in physics, engineering, economics, statistics, and more. Most importantly, calculus allows for the evaluation of the exact instantaneous change and accumulation, providing a precise understanding of dynamically changing systems.

Continuous Functions

A continuous function is one in which small changes in the input produce small changes in the output. In other words, there are no abrupt jumps, holes, or breaks in the graph of a continuous function. A function is continuous over a certain interval if it meets three conditions at any point within that interval:

1. The function is defined at the point.
2. The limit exists at that point.
3. The function's value equals the limit at that point.

Continuous functions are extremely important in calculus because they ensure that the limit of a function at any point is the same as the function's value at that point, as seen in the problem concerning the draining cylindrical tank.

Real-Life Applications of Calculus

Calculus is far more than an abstract mathematical discipline; it has countless real-life applications. Engineers use calculus to design curves that minimize surface area or maximize volume. Economists use it to model growth trends and optimize resources. Scientists apply calculus to test hypotheses and predict changes in complex systems such as weather patterns and biological processes.

• Physics: Calculus helps in studying motion and forces.
• Engineering: It is used for designing and understanding the behavior of structures.
• Economics: Calculus assists in cost optimization and economic forecasts.

In the example of the draining tank, calculus offers a way to predict the emptying process, allowing for precise measurements in fluid dynamics, a fundamental concern for various industries from manufacturing to environmental management.

Limit Evaluation

Limit evaluation is the process of finding the value that a function approaches as the input approaches a certain value. This is fundamental to calculus because it deals with quantities that are changing and not static. The technique of substituting the value directly, as seen in the draining tank problem, is applicable when the function is continuous. However, when the function isn't continuous at that specific input, alternative methods such as factoring, rationalization, or applying L'Hôpital's rule might be used.

Direct Substitution

If a function is continuous at a certain point, we can find the limit by simply plugging in the value of the variable.

Factoring

When a function has a common factor in the numerator and denominator, factoring can be used to simplify the function and evaluate the limit.

Rationalization

For functions involving roots, rationalization helps remove radicals from the denominator or numerator to make the limit easier to evaluate.