By the Pythagorean Inequality Theorem, a triangle is acute if \(a2+b2>c2a\) where c is the longest side. So, we can write

\(\displaystyle{9}^{{2}}+{12}^{{2}}{>}{c}^{{2}}\)

\(\displaystyle{81}+{144}{>}{c}^{{2}}\)

\(\displaystyle{225}{>}{c}^{{2}}\)

\(\displaystyle{15}{>}{c}\)

This means that the longest side must be less than 15 inches. Hence, the e greatest possible whole number that can be the length of the longest side of the triangle is 14 inches.