A billboard designer has decided that a sign should have 3 ft margins at the top and bottom and 5 ft margins on the left and right sides furthermore the billboard should have a total area of 900ft^2 (including the margins) if x denotes the width in feet of the billboard find the function in the variable x giving the area of the printed region of the billboard

Sagot :

Answer:

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Step-by-step explanation:

Answer:

The width of billboard is "[x]" and the height of billboard is "[y"]. If total area of billboard is [tex]9000 ft^2[/tex] then [tex]9000=xy[/tex]

Step-by-step explanation:

• The total width of billboard is [x]. Therefore the width of printed area will be (x-10) by excluding margin of left and right side.

• The total height of billboard is [y]. Therefore the height of printed area will be [(y-6)]  by excluding the margin of top and bottom from the total height.

• To find the printed area of billboard calculations are given below:

[tex]& 9000=xy[/tex]

[tex]& y=\frac{9000}{x} \\ & A=(x-10)(y-6) \\ & A=xy-6x-10y+60 \\ & A=x\left( \frac{9000}{x} \right)-6x-10\left( \frac{9000}{x} \right)+60 \\ & A=9060-6x-\frac{9000}{x} \\[/tex]

On taking the first order derivative of A

[tex]\[A'=-6+\left( \frac{90000}{{{x}^{2}}} \right)\][/tex]

[tex]& \left( \frac{90000}{{{x}^{2}}} \right)-6=0 \\ & 6{{x}^{2}}=90000 \\ & x=\sqrt{15000} \\ & y=\frac{9000}{x}=\frac{90000}{\sqrt{15000}}=10\sqrt{150} \\[/tex]

• Hence [tex]\[x=10\sqrt{150}\][/tex] and [tex]\[y=\frac{900}{\sqrt{150}}\][/tex]

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