UpStudy Free Solution:

To determine how far Mr. Pearson and his son Connor can throw the football from one corner of the acre to the opposite corner, we need to calculate the diagonal of the square plot.

First, we need to find the side length of the square plot of land. Since the area of a square is given by the formula \(\text { side} ^ 2\), we can set up the equation:

\[\text { side} ^ 2 = 43,560 \space \text { sq ft} \]

Taking the square root of both sides:

\[\text { side} = \sqrt { 43,560} \approx 208.71 \space \text { ft} \]

Next, we use the Pythagorean theorem to find the diagonal of the square. For a square, the diagonal \(d\) can be calculated using:

\[d = \sqrt { \text { side} ^ 2 + \text { side} ^ 2} = \sqrt { 2 \times \text { side} ^ 2} = \text { side} \times \sqrt { 2} \]

Substituting the side length:

\[d = 208.71 \times \sqrt { 2} \approx 208.71 \times 1.414 \approx 295.16 \space \text { ft} \]

Therefore, Mr. Pearson and his son Connor can throw the football approximately 295.16 feet from one corner of the acre to the opposite corner.

Supplemental Knowledge

Calculating the diagonal of a square plot involves understanding basic geometric principles and applying the Pythagorean theorem. Here’s a deeper dive into the concepts used:

Key Concepts:

1. Area of a Square:

- The area (\(A\)) of a square is given by \(A = \text { side} ^ 2\).

- Given an area, the side length (\(\text { side} \)) can be found by taking the square root: \(\text { side} = \sqrt { A} \).

2. Pythagorean Theorem:

- For any right-angled triangle, the relationship between the lengths of the sides is given by \(a^ 2 + b^ 2 = c^ 2\), where \(c\) is the hypotenuse.

- In a square, each diagonal forms two right-angled triangles with sides equal to the side length of the square.

3. Diagonal of a Square:

- The diagonal (\(d\)) splits the square into two congruent right-angled triangles.

- Using the Pythagorean theorem for these triangles, \(d = \sqrt { \text { side} ^ 2 + \text { side} ^ 2} = \sqrt { 2 \times \text { side} ^ 2} = \text { side} \times \sqrt { 2} \).

Additional Example:

Consider another example where you have a rectangular plot with dimensions 50 feet by 30 feet. To find its diagonal, apply:

\[d = \sqrt { 50^ 2 + 30^ 2} = \sqrt { 2500 + 900} = \sqrt { 3400} \approx 58.31\space \text { ft} .\]

This same principle applies to any rectangular or square plot.

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