Problem 1 - Calculus: Early Transcendentals

Question

A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume \(V\) of water remaining in the tank (in gallons) after \(t\) minutes. $$ \begin{array}{|c|c|c|c|c|c|c|} \hline t(\mathrm{~min}) & 5 & 10 & 15 & 20 & 25 & 30 \\ \hline V(\mathrm{gal}) & 694 & 444 & 250 & 111 & 28 & 0 \\ \hline \end{array} $$ (a) If \(P\) is the point (15,250) on the graph of \(V\), find the slopes of the secant lines \(P Q\) when \(Q\) is the point on the graph with \(t=5,10,20,25,\) and 30 (b) Estimate the slope of the tangent line at \(P\) by averaging the slopes of two secant lines. (c) Use a graph of the function to estimate the slope of the tangent line at \(P\). (This slope represents the rate at which the water is flowing from the tank after 15 minutes.)

Step-by-Step Solution

Answer

(a) Slopes: -44.4, -38.8, -27.8, -22.2, -16.67. (b) Estimated tangent slope: -33.3. (c) Visually: about -33.3.

Step 1: Find the Slope of Secant Line PQ for t=5

First, identify the points \((t, V)\) for \(Q\) at \(t = 5\). The coordinates are \(Q(5, 694)\). For each secant line \(PQ\), use the formula for the slope: \(m = \frac{V_2 - V_1}{t_2 - t_1}\).For \(t = 5\), the slope \(m\) is:\[m = \frac{250 - 694}{15 - 5} = \frac{-444}{10} = -44.4\]

Step 2: Find the Slope of Secant Line PQ for t=10

Next, determine the coordinates for the point \(Q(10, 444)\).For \(t = 10\), the slope \(m\) is:\[m = \frac{250 - 444}{15 - 10} = \frac{-194}{5} = -38.8\]

Step 3: Find the Slope of Secant Line PQ for t=20

Next, calculate the slope using the coordinates for \(Q(20, 111)\).For \(t = 20\), the slope \(m\) is:\[m = \frac{250 - 111}{15 - 20} = \frac{139}{-5} = -27.8\]

Step 4: Find the Slope of Secant Line PQ for t=25

Next, calculate for the point \(Q(25, 28)\).For \(t = 25\), the slope \(m\) is:\[m = \frac{250 - 28}{15 - 25} = \frac{222}{-10} = -22.2\]

Step 5: Find the Slope of Secant Line PQ for t=30

Finally, find the slope using the coordinates for \(Q(30, 0)\).For \(t = 30\), the slope \(m\) is:\[m = \frac{250 - 0}{15 - 30} = \frac{250}{-15} = -16.67\]

Step 6: Estimate the Slope of the Tangent Line at P by Averaging

To estimate the tangent slope at \(P\), average the slopes of two appropriate secant lines, commonly the ones immediately before and after \(P\). Let's use \(t = 10\) and \(t = 20\):\[\text{Average slope} = \frac{-38.8 + (-27.8)}{2} = \frac{-66.6}{2} = -33.3\]

Step 7: Estimate Using Graph (General Guidance)

By sketching or examining a graph, approximate the slope of the tangent at point \(P(15, 250)\). The slope can visually be estimated, commonly around the average of slopes determined earlier, confirming it approximately matches \(-33.3\) gallons per minute. This graph illustrates the behavior of water draining over time.

Key Concepts

Secant Line

A secant line is simply a straight line that cuts through a curve at two points. In calculus, the secant line helps us understand how much a function changes over certain intervals. By working with secant lines, we can anticipate how the slope of a curve behaves over larger segments of the graph.

To find the slope of a secant line between two points on a curve, use the formula: \[ m = \frac{V_2 - V_1}{t_2 - t_1} \]This equation calculates the change in the vertical values, often called the "rise," divided by the change in horizontal values, known as the "run."

• "Rise" is the change in values of the function's outputs, in this scenario, the water volume.
• "Run" is the change in the input values or time.

Using the data, you calculate different secant slopes between point \(P\)(15, 250) and various \(Q\) coordinates. Doing this gives insight into how steep or flat the function graph is at different intervals.

Tangent Line

The tangent line is closely related to the secant line but is more specific. Imagine zooming in on the part of the curve that passes through a single point. At this small scale, a tangent line just barely touches the curve at one point without intersecting fully on more than that one spot.

The tangent line's slope tells us the "instant" rate of change of the function at a particular point. It's the equivalent of saying, "how fast is this curve moving right here?" This is incredibly useful in calculus for understanding motion and changing processes.

• The tangent line helps isolate the exact rate of change at a specific moment.
• It is derived by approximating or finding the limit of the slope of secant lines as two points become infinitesimally close.

This idea aids in estimating derivatives, offering a glimpse into more precise changes over time, such as how quickly a variable like water drains after a specific interval.

Slope Estimation

Estimating the slope of a tangent line often requires combining the information we get from secant lines. If we want a good guess at the slope at a particular point, we might average the slopes of two secant lines that border that point closely.

In practice:

• Choose two points that closely surround your target point \(t=15\).
• Calculate the slope of the secant lines for these points.
• Averaging these slopes gives a value that closely represents the tangent slope.

This process brings us closer to a hover level understanding of the behavior of our function. This simple technique acts as a bridge to more complex calculus concepts like taking derivatives but remains grounded in fundamental arithmetic that retains its real-world practicality. It's like finding the average speed of a journey by considering speeds at various checkpoints.

Rate of Change

The rate of change is a crucial concept in calculus, representing how fast variables in a function change in relation to each other. High rates can show rapid changes while low rates indicate more gradual changes.

In the context of a draining tank, the rate of change tells us how quickly the water level decreases over time. We can calculate this using both tangent and secant slopes.

• Secant lines give a broader look at changes over intervals, showing overall water loss.
• Tangent lines zoom into specific time points to give a clearer picture of instantaneous flow rates.

Understanding the rate of change is essential. It helps us to respond to changing conditions proactively. Whether managing resources or predicting future needs, the rate provides insight into efficiency and necessary adjustments. For instance, knowing the flow rate at 15 minutes can help determine if intervention is needed right now or can wait.