Problem 24 - Algebra 2

Question

A center-pivot irrigation system can water from 1 to 130 acres of crop land. The length \(\ell\) in feet of rotating pipe needed to irrigate \(A\) acres is given by the function \(\ell=117.75 \sqrt{A}\). a. Graph the equation on your calculator. Make a sketch of the graph. b. Find the lengths of pipe needed to irrigate \(40,80,\) and 130 acres.

Step-by-Step Solution

Answer

To irrigate 40, 80, and 130 acres, you would need approximately 745.5 feet, 1053.3 feet, and 1326.3 feet of pipe, respectively.

Step 1: Graphing the Equation

Use the graphing function on your calculator to plot \(\ell=117.75 \sqrt{A}\). In the calculator, x-values can substitute for \(A\) and y-values for \(\ell\). Ensure the values range from 1 to 130 on your x-axis to cover the acreage mentioned in the question. The curve you will get represents the change in length of the pipe as the area increases.

Step 2: Finding the Lengths at Specific Areas

First, substitute \(A = 40\) into the function to get \(\ell = 117.75 \sqrt{40}\). Using a calculator, we find this is approximately \(\ell \approx 745.5\) feet. Next, substitute \(A = 80\) into the function to get \(\ell = 117.75 \sqrt{80}\), this gives \(\ell \approx 1053.3\) feet. Lastly, substitute \(A = 130\) into the function to get \(\ell = 117.75 \sqrt{130}\), yielding \(\ell \approx 1326.3\) feet.These lengths tell us how long the pipe should be to irrigate the corresponding area.

Key Concepts

Functions

Functions in mathematics are essential as they describe a relationship between a set of inputs and a set of possible outputs. In this exercise, the function is \( \ell = 117.75 \sqrt{A} \), and it relates the length of the irrigation pipe \( \ell \) to the area of the land \( A \) that needs to be watered.
Functions can be thought of as machines: you input a number, the function does something to it, and you get an output.
For example:

• Input: \( A = 40 \)
• Process: The function calculates \( \ell = 117.75 \sqrt{40} \)
• Output: \( \ell \approx 745.5 \)

Functions are critical in mathematical modeling for translating real-world scenarios, like irrigation, into mathematical language.This allows us to predict outcomes and make decisions based on different inputs.

Graphing Equations

Graphing equations is a powerful tool to visualize the relationships described by a function. In our given problem, we are graphing the function \( \ell = 117.75 \sqrt{A} \). This helps to understand how the length of the irrigation pipe changes with different acreages.
To begin, consider \( A \) as the x-axis (independent variable) and \( \ell \) as the y-axis (dependent variable).
Using a graphing calculator, input a series of values for \( A \) within the domain, which ranges from 1 to 130.
Plot the resulting values of \( \ell \) to create a curve.
This graph visually demonstrates:

• The larger the area, the longer the pipe needs to be.
• The shape of the curve represents a square root function, which increases at a decreasing rate.

Seeing this curve allows us to quickly estimate the required pipe length for any given acreage.

Square Roots

A square root is a value that, when multiplied by itself, gives the original number. In the function \( \ell = 117.75 \sqrt{A} \), the square root of \( A \) is used to calculate the pipe length.
Understanding square roots is important because they often appear in functions modeling real-world scenarios.
For instance, knowing that \( \sqrt{40} \approx 6.32 \) helps find that \( \ell \approx 117.75 \times 6.32 = 745.5 \).
Square roots can sometimes seem tricky, but they essentially reverse the process of squaring a number:

• If \( x^2 = 16 \), then \( x = \pm 4 \)
• Square roots of non-perfect squares can be approximated using calculators.

The use of square roots in this problem demonstrates how seemingly abstract mathematical concepts can be applied in practical solutions, like measuring pipe length.

Mathematical Modeling

Mathematical modeling involves using equations to represent real-world situations. It's an essential skill in mathematics and practical applications, like engineering or environmental science.
In our problem, the function \( \ell = 117.75 \sqrt{A} \) models the relationship between the area irrigated and the pipe length required.
Mathematical models help:

• Predict future outcomes, such as how much pipe is needed for a new field.
• Optimize resources by determining the most efficient lengths for different areas.
• Better understand complex systems through simplified representations.

By setting up an equation based on real-world data, you can solve for unknowns and make informed decisions.
Mathematical modeling is not only about solving equations, but also interpreting the results within the context of the scenario.