Question

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phyllis bought 21 feet of wood to frame a window. if she makes a triangular window, what is the greatest area the window can have?
for a given perimeter, the triangle with the largest area is triangle. using this type of triangle, the sides would measure . so, the greatest area the window could be is about square feet.

Explanation:

Step1: Identify optimal triangle type

For fixed perimeter, equilateral triangle has maximum area.

Step2: Calculate side length

Total perimeter = 21 ft. For equilateral triangle, side length $s = \frac{21}{3} = 7$ ft.

Step3: Compute triangle area

Use equilateral area formula: $A = \frac{\sqrt{3}}{4}s^2$
Substitute $s=7$:
$A = \frac{\sqrt{3}}{4} \times 7^2 = \frac{\sqrt{3}}{4} \times 49 \approx 21.22$

Answer:

For a given perimeter, the triangle with the largest area is equilateral triangle. Using this type of triangle, the sides would measure 7 feet each. So, the greatest area the window could be is about 21.22 square feet.