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Answered question

2020-12-22

What are the measurements (width and length) for the maximum A of a rectangular with P=1000?

Answer & Explanation

Adnaan Franks

Skilled2020-12-23Added 92 answers

Let the length and width be x and y
P=2x+2y
Substitute P=1000
1000=2x+2y
1000-2x=2y
500-x=y
A=xy
Substitute:
A=x(500-x)
$A = 500 x - x^{2}$
A quadratic function ( $a x^{2} \pm b x \pm c$ ) has it's maximum at its vertex. Since a=-1<0, A has a maximum. The x minimum is $x = - \frac{b}{2 a}$ . Since a=-1 and b=500:
$x = - \frac{500}{2 (- 1)} = 250$
Substitute this into y=500-x:
y=500-250=250
Max length and width are:
250, 250

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