We'll let the red star represent the acute angle in each case.
Notice that, as the acute angle approaches 90°, the length of the diagonal increases.

So the greatest value of the length of the diagonal opposite the acute angle will occur when the acute angle is as close as possible to 90° (e.g., 89.99999999999999999°)

If the angle is 90°, then we can apply the Pythagorean theorem to see that that the length of the diagonal \(= \sqrt{221}\)
So, if the acute angle were 89.99999999999999999°, the length of the diagonal would be a teeny tiny bit less than \(\sqrt{221}\)

We already know that:
\(\sqrt{196}=14\)
and
\(\sqrt{225}=15\)

So, \(\sqrt{221}\) must be BETWEEN 14 and 15

Since the length of the diagonal must be an integer, the greatest possible length is 14

Answer: 14

Cheers,
Brent

Thank you Sir for the explanation provided.

Regards

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