An equilateral triangle has an area of \(36\sqrt{3}\) square centimeters. If each side of the triangle is decreased by 4 cm, by how many square centimeters is the area reduced?
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- The shortest altitude corresponds to the longest side, which is \(c = 15\):
- h_{\text{min}} = \frac{2 \times 84}{15} = \frac{168}{15} = \frac{56}{5} = 11.2
- Thus, the length of the shortest altitude is \(\boxed{\frac{56}{5}}\) units.
- The area \(A\) of an equilateral triangle with side \(s\) is:
- Given \(A = 36\sqrt{3}\), solve for \(s\):
- \frac{\sqrt{3}}{4} s^2 = 36\sqrt{3} \implies \frac{s^2}{4} = 36 \implies s^2 = 144 \implies s = 12 \text{ cm}
