Henry Ernest Dudeney/Modern Puzzles/130 - Mr. Grindle's Garden/Solution/Proof 1

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Modern Puzzles by Henry Ernest Dudeney: $130$

Mr. Grindle's Garden
"My neighbour," said Mr. Grindle, "generously offered me, for a garden,
as much land as I could enclose with four straight walls measuring $7$, $8$, $9$ and $10$ rods in length respectively."
"And what was the largest area you were able to enclose?" asked his friend.
Perhaps the reader can discover Mr. Grindle's correct answer.


Just under $71$ square rods.


Let $\AA$ square rods be the area of the garden.

We are given that $\AA$ is the greatest possible for the given sides.

We also have that the sides of the garden are in arithmetic sequence.


\(\ds \AA\) \(=\) \(\ds \sqrt {7 \times 8 \times 9 \times 10}\) Greatest Area of Quadrilateral with Sides in Arithmetic Sequence
\(\ds \) \(=\) \(\ds \sqrt {5040}\)
\(\ds \) \(\approx\) \(\ds 70.99\)