### Instruction: An equilateral triangle has an area of $64\sqrt{3}$ $\text{cm}^2$. If each side of the triangle is decreased by 4 cm, by how many square centimeters is the area decreased?### Response: To solve this problem, I need to find the original side length of the triangle, and then the new side length after decreasing it by 4 cm.
Then I can use the formula for the area of an equilateral triangle to find the difference in area.
The formula for the area of an equilateral triangle with side length s is $\frac{s^2\sqrt{3}}{4}$.
So, if the area is $64\sqrt{3}$ $\text{cm}^2$, I can set up an equation and solve for s:

\[\frac{s^2\sqrt{3}}{4} = 64\sqrt{3}\]
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Multiplying both sides by 4 and dividing by $\sqrt{3}$, I get:

\[s^2 = 256\]
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Taking the square root of both sides, I get:

\[s = 16\]
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So the original side length of the triangle is 16 cm.
If I decrease it by 4 cm, the new side length is 12 cm.
To find the new area, I plug in 12 for s in the formula:

\[\frac{12^2\sqrt{3}}{4} = 36\sqrt{3}\]
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