Answered

Number of Fencing Panels Required for Different Square-Shaped Areas




Area ( $ft .{ }^2$ )		Number of Fencing Panels

25		4

100		8

225		12

400		16

625		20

900		24






The table shows the number of fencing panels needed to surround a given square-shaped area. Use technology to create a square root function that models this data.


AThe number of fencing panels needed as a function of area is modeled by the function $p(x)=1.25 \sqrt{x}$.
BThe number of fencing panels needed as a function of area is modeled by the function $p(x)=0.8 \sqrt{x}$.
CThe number of fencing panels needed as a function of area is modeled by the function $p(x)=\sqrt{x}-2$.
DThe number of fencing panels needed as a function of area is modeled by the function $p(x)=\sqrt{x}-1$.

Verified Answer :

  • We analyzed the provided data table for area (\(x\)) and fencing panels (\(p\)).
  • We tested each of the four proposed functions against the data points, verifying if the output \(p\) matches the table.
  • The function \(p(x) = 0.8\sqrt{x}\) yielded the correct results for all data points given (e.g., at \(x=100\), \(p=8\); at \(x=900\), \(p=24\)).
  • Thus, the model is \(p(x) = 0.8\sqrt{x}\).
  • Final Answer: \(\boxed{p(x) = 0.8\sqrt{x}}\).

Explanation

  1. Analysis of the Data and Candidate Functions

The problem asks to identify which of the provided square root functions correctly models the relationship between the area of a square \(x\) (in square feet) and the number of fencing panels \(p\) required to surround it. The data points from the table are: \((x, p) = (25, 4), (100, 8), (225, 12), (400, 16), (625, 20), (900, 24)\). We need to test the given candidate functions:
1) \(p(x) = 1.25\sqrt{x}\)
2) \(p(x) = 0.8\sqrt{x}\)
3) \(p(x) = \sqrt{x} - 2\)
4) \(p(x) = \sqrt{x} - 1\)

  1. Testing the Functions with Data Points

To determine the correct model, we test each function using the provided data points.
For \(x = 25\):
1) \(1.25\sqrt{25} = 1.25 \times 5 = 6.25 eq 4\)
2) \(0.8\sqrt{25} = 0.8 \times 5 = 4\) (Matches)
3) \(\sqrt{25} - 2 = 5 - 2 = 3 eq 4\)
4) \(\sqrt{25} - 1 = 5 - 1 = 4\) (Matches)

Since both 2 and 4 match at \(x=25\), we must check another point. Let's use \(x = 100\):
2) \(0.8\sqrt{100} = 0.8 \times 10 = 8\) (Matches)
4) \(\sqrt{100} - 1 = 10 - 1 = 9 eq 8\)

For \(x = 225\):
2) \(0.8\sqrt{225} = 0.8 \times 15 = 12\) (Matches)

  1. Conclusion and Final Answer

We have confirmed that the function \(p(x) = 0.8\sqrt{x}\) consistently produces the correct number of panels for the given areas. We can verify it for the remaining points as well:

  • For \(x = 400\): \(0.8\sqrt{400} = 0.8 \times 20 = 16\)
  • For \(x = 625\): \(0.8\sqrt{625} = 0.8 \times 25 = 20\)
  • For \(x = 900\): \(0.8\sqrt{900} = 0.8 \times 30 = 24\)

All values match the table perfectly. Therefore, the function that models the relationship is \(p(x) = 0.8\sqrt{x}\).

Examples

This type of modeling is essential for construction and landscaping projects. If you are a gardener planning a square garden, you need to calculate how many fence panels to purchase based on the square footage you plan to enclose. By developing a mathematical function from observed data, you can quickly estimate costs and materials for any project size, saving both time and resources in your planning process.

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