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 rig

Question

3 years ago

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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 rig

ht 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

Mathematics

2 answers:

elixir [45] · Answer 1 · 2022-07-24T22:44:04+03:00

Answer:

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

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:

$& 9000=xy$

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

On taking the first order derivative of A

$A'=-6+\left( \frac{90000}{{{x}^{2}}} \right)$

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