Answer to Question #349932 in Calculus for Mark

Question #349932

. A poster must have 32 sq.in. of printed materials with margins of 4 inches each at the top and 2 inches at each side. Find the dimensions of the whole poster if its area is maximum.


1
Expert's answer
2022-06-13T23:27:15-0400

Let x=x= the length of printed matter, y=y=the width of printed matter.

Given xy=32.xy=32.

The length of a poster will be x+2(2),x+2(2), the width of a poster will be y+4.y+4.

The area of a poster is A=(x+4)(y+4).A=(x+4)(y+4).



xy=32=>y=32xxy=32=>y=\dfrac{32}{x}

Substitute



A=A(x)=(x+4)(32x+4),x>0A=A(x)=(x+4)(\dfrac{32}{x}+4), x>0A(x)=32+4x+128x+16A(x)=32+4x+\dfrac{128}{x}+16

Find the derivative with respect to xx



A(x)=(48+4x+128x)=4128x2A'(x)=(48+4x+\dfrac{128}{x})'=4-\dfrac{128}{x^2}

Find the critical number(s)



A(x)=0=>4128x2=0A'(x)=0=>4-\dfrac{128}{x^2}=0x2=32x^2=32x1=42,x2=42x_1=-4\sqrt{2}, x_2=4\sqrt{2}

If 0<x<42,A(x)>0,A(x)0<x<4\sqrt{2}, A'(x)>0, A(x) increases.


If x>42,A(x)<0,A(x)x>4\sqrt{2}, A'(x)<0, A(x) decreases.


The function A(x)A(x) has a local maximum at x=42.x=4\sqrt{2}.

Since the function A(x)A(x) has the only extremum for x>0,x>0, then the function A(x)A(x) has the absolute maximum at x=42x=4\sqrt{2} for x>0.x>0.



y=3242=42y=\dfrac{32}{4\sqrt{2}}=4\sqrt{2}

The poster is (42+4) inches×(42+4) inches.(4\sqrt{2}+4)\ inches\times (4\sqrt{2}+4)\ inches.



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