Answer to Question #343546 in Calculus for yousii

Question #343546

A poster is to have an area of 630 cm2 with 2.5 cm margins at the bottom and sides and a 5 cm margin at the top. Find the exact dimensions (in cm) that will give the largest printed area.



width : cm



height: cm

1
Expert's answer
2022-05-24T09:54:11-0400

Let x=x= width of the poster, let y=y= height of the poster.

Then


xy=630xy=630

The largest printed area will be


A=(x2.52.5)(y2.55)A=(x-2.5-2.5)(y-2.5-5)

Substitute

A=A(x)=(x5)(630x7.5)A=A(x)=(x-5)(\dfrac{630}{x}-7.5)

=6307.5x3150x+37.5=630-7.5x-\dfrac{3150}{x}+37.5

=667.57.5x3150x,5x84=667.5-7.5x-\dfrac{3150}{x}, 5\le x\le 84

Differentiate wih respect to xx


A=7.5+3150x2A'=-7.5+\dfrac{3150}{x^2}

Find the critical number(s)


A=0=>7.5+3150x2=0A'=0=>-7.5+\dfrac{3150}{x^2}=0

x=±420x=\pm\sqrt{420}

Since 5x84,5\le x\le 84, we take x=420x=\sqrt{420}

If 5x<420,A>0,A5\le x<\sqrt{420}, A'>0, A increases.

If 420<x420,A<0,A\sqrt{420}<x\le\sqrt{420}, A'<0, A decreases.

The function AA has the absolute maximum for 5x845\le x\le 84 at x=420.x=\sqrt{420}.


y=630420=945y=\dfrac{630}{\sqrt{420}}=\sqrt{945}

width : 420\sqrt{420} cm


height: 945\sqrt{945} cm


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