If a sector of maximum area is made with a wire of length 40 cm, then the area (in sq cms) of that sector is
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For a sector with a fixed perimeter \( P \), the maximum possible area is always achieved when the arc length equals twice the radius (\( l = 2r \)). This means the maximum area is simply given by the elegant formula: \( A = \frac{P^2}{16} \).
Step 1: Write the perimeter relation. A sector of radius $r$ and arc length $l$ has perimeter $2r+l$. The wire is 40 cm, so \[ 2r+l=40\implies l=40-2r \] Step 2: Write the area. The area of a sector is $A=\frac{1}{2}rl$. Step 3: Make the area depend on $r$ only. \[ A=\frac{1}{2}r(40-2r)=20r-r^2 \] Step 4: Differentiate and set to zero. \[ \frac{dA}{dr}=20-2r=0\implies r=10 \] Step 5: Confirm it is a maximum. The second derivative is $\frac{d^2A}{dr^2}=-2<0$, so $r=10$ gives the largest area. Step 6: Find the maximum area. \[ A_{\max}=20(10)-(10)^2=200-100=100 \] \[ \boxed{100\text{ sq cm}} \]