- Analyze the provided data points \((10, 900), (20, 1600), (30, 2100)\).
- Substitute these widths into the candidate quadratic equations to test for the corresponding area values.
- Confirm that the function \(A(w) = -w^2 + 100w\) yields the correct area for all given data points.
- Conclude that the correct equation is option (d), \(\boxed{d}\).
Explanation
- Data Analysis and Objectives
The problem asks us to find the quadratic function \(A(w)\) that models the relationship between the width \(w\) of a rectangle (in feet) and its area \(A\) (in square feet). We are given three data points: \((w_1, A_1) = (10, 900)\), \((w_2, A_2) = (20, 1600)\), and \((w_3, A_3) = (30, 2100)\). We need to determine which of the provided candidate functions accurately represents these points. The candidate equations are:
A\(A(w) = -w^2 + 200w\)
B\(A(w) = w^2 + 40w\)
C\(A(w) = w^2 + 90w\)
D\(A(w) = -w^2 + 100w\)
- Testing Candidate Equations
To identify the correct equation, we will test each candidate by substituting the given widths \(w\) and checking if the resulting area matches the data points in the table. We begin by testing the first point \((w=10, A=900)\):
- For (a) \(A(10) = -(10)^2 + 200(10) = -100 + 2000 = 1900\). (Incorrect)
- For (b) \(A(10) = (10)^2 + 40(10) = 100 + 400 = 500\). (Incorrect)
- For (c) \(A(10) = (10)^2 + 90(10) = 100 + 900 = 1000\). (Incorrect)
- For (d) \(A(10) = -(10)^2 + 100(10) = -100 + 1000 = 900\). (Matches!)
Since only option (d) satisfies the first data point, it is the most likely candidate. We proceed by verifying option (d) with the remaining data points.
- Verification of the Correct Model
Now, we verify candidate (d), \(A(w) = -w^2 + 100w\), with the other provided points to ensure consistency:
- Testing point \((w=20, A=1600)\):
\(A(20) = -(20)^2 + 100(20) = -400 + 2000 = 1600\). (Matches!)
- Testing point \((w=30, A=2100)\):
\(A(30) = -(30)^2 + 100(30) = -900 + 3000 = 2100\). (Matches!)
All three points are consistent with the function \(A(w) = -w^2 + 100w\). Thus, this is the correct quadratic equation representing the relationship.
- Final Conclusion
Based on our testing, the quadratic function \(A(w) = -w^2 + 100w\) correctly models the area of the rectangle for all given widths. Therefore, option (d) is the correct choice. The final answer is \(\boxed{d}\).
Examples
In real-world applications like agriculture or land management, farmers often need to calculate the maximum area they can enclose with a fixed length of fencing. Understanding how the area of a rectangle changes with its width allows a rancher to determine the optimal dimensions to maximize space for livestock. This quadratic relationship is a foundational concept in resource optimization and geometry.