Problem 80 - Algebra and Trigonometry Real Mathematics, Real People

Question

The force \(F\) (in tons) of water against the face of a dam is estimated by the function $$F(y)=149.76 \sqrt{10} y^{5 / 2}$$ where \(y\) is the depth of the water (in feet). (a) Complete the table. What can you conclude? $$\begin{array}{|l|l|l|l|l|l|} \hline y & 5 & 10 & 20 & 30 & 40 \\ \hline F(y) & & & & & \\ \hline \end{array}$$ (b) Use a graphing utility to graph the function. Describe your viewing window. (c) Use the table to approximate the depth at which the force against the dam is 1,000,000 tons. Verify your answer graphically. How could you find a better estimate?

Step-by-Step Solution

Answer

The force reaches 1,000,000 tons between a depth of 10 and 20 feet. This can be observed both graphically and using the table. A more precise estimate could be acquired using more sophisticated mathematical methods such as quadratic or cubic interpolation.

Step 1: Evaluate the function for given values

First, evaluate the function \(F(y) = 149.76 \sqrt{10} y^{5 / 2}\) for values of \(y\) as 5, 10, 20, 30, and 40. This step involves substituting these values in the function and calculating the corresponding output values. Using a calculator, the function F(y) values for given y values can be found.

Step 2: Create the table

The given table can be completed by inserting calculated values of \(F(y)\). The table after inserting these values will look like this: \n\(\begin{array}{|l|l|l|l|l|l|} \hline y & 5 & 10 & 20 & 30 & 40 \ \hline F(y) & 11910 & 149760 & 1198080 & 4046748 & 9589056 \ \hline \end{array}\)From this table, we can observe that as depth (y) increases, the force against the dam (F(y)) also increases.

Step 3: Graph the function

Use a graphing utility to plot \(F(y)\) as a function of \(y\). The graph will be of a curve that rapidly increases as \(y\) increases. The viewing window can be adjusted to fit all relevant points and provide a detailed view of the graph.

Step 4: Approximate the depth at which the force against the dam is 1,000,000 tons

Examine the table and use it to interpolate the depth (y value) at which the force against the dam is 1,000,000 tons. This is done by finding two points between which the value 1,000,000 lies and estimating an intermediate value. The estimated value can be verified by inspecting the graph. A more accurate estimate could be found by using methods such as quadratic or cubic interpolation, or by using a larger table with smaller increments for y. From the table, it can be seen that a force of 1,000,000 tons is reached between a depth of 10 and 20 feet. Therefore, the answer is within this interval. By inspecting the graph, this can be visually confirmed.

Key Concepts

Function Evaluation

Understanding and evaluating functions correctly is crucial in numerous real-world problems, such as determining water force against a dam. The given function, \(F(y) = 149.76 \sqrt{10} y^{5/2}\), expresses force as a function of depth \(y\). Let's understand what evaluating this function practically means: we need to calculate the force for specific depths. To evaluate this:

• Select a set of values for \(y\) (such as 5, 10, 20, 30, and 40 feet in this case).
• Substitute these values into the function and calculate \(F(y)\) using a calculator.

For example, substituting \(y = 5\) into the function gives us \(F(5)\). This involves calculating the expression inside the square root and then multiplying by the coefficient. By repeating this for all selected values of \(y\), we can construct a table showing how the force changes with increasing water depth. This is key to visualizing and understanding the force's behavior relative to depth.

Graphing Utility

Graphing utilities are incredibly useful tools that allow us to visualize mathematical functions to better understand how they behave. In this exercise, we use a graphing utility to visually represent the function \(F(y) = 149.76 \sqrt{10} y^{5/2}\), showing the relationship between depth \(y\) and water force \(F\). To effectively use a graphing utility:

• First, input the function correctly into the utility.
• Adjust the viewing window settings to ensure all relevant parts of the graph are visible. For example, choose an appropriate range for \(y\) (like 0 to 40 feet) and force \(F\) to see the curve clearly.
• Observe how the graph increases, confirming the relationship where greater depth results in stronger forces against the dam.

This graphical representation is not only visually descriptive, but it also helps in comparisons and predictions, such as verifying specific force values against depth, or spotting trends that are not obvious from the function alone.

Depth Interpolation

Interpolation is a technique used to estimate unknown values that fall within the range of known data points. In this problem, we utilize interpolation to estimate the water depth for a specific force value against the dam. The table shows calculated forces for given depths. We want to find the depth for which the force equals 1,000,000 tons. With depth interpolation:

• Identify the interval in the table where this force value lies. For instance, a force of 1,000,000 tons is between the depths of 10 and 20 feet.
• Use a simple linear interpolation method: imagine a line connecting the points (10, 149760) and (20, 1198080), and find the \(y\) value where the line crosses 1,000,000 on the force axis.
• Alternatively, refine this estimate using more advanced interpolation methods, such as quadratic or cubic interpolation, for greater accuracy.

Visual verification can be done using the graph, checking where the force equals 1,000,000 tons. This combination of interpolation and graphical confirmation provides a comprehensive approach for solving the problem.