If I was trying to design a printed billboard using a minimal area of plywood. The printed area must be $2000$ sq ft.Here are the margins. side margins $10$ ft top margin $8$ ft bottom margin $6$ ft How can I find the optimal, width, height, and total area?


You want to minimize area. Let's call the printed area of dimension $x\times y$ where

$x$:= width of printed area, $y$: = height of printed area.

$xy = 2000\tag{1 printed Area }$

Now, we have to compute the width of the billboard with side margins of 10 feet: $\;w = x + 2\cdot 10 = x + 20$

And the height of the billboard being: $\;y + 8 + 6 = y + 14$

We want to minimize the total area $A$ of the billboard, with $$\;A = (x+ 20)(y+ 14) = 14x + xy + 20y + 280\tag{Billboard Area}$$

Using $(1)$: $y = \dfrac {2000}{x}$

Substituting into the equation for the area of the billboard gives us $$A = 14x + 2000 + 20\cdot \frac{2000}x + 280 = 14x + \dfrac{40000}{x} + 2280\tag{A}$$

Now, find the derivative of $A$, $A'$ with respect to $x$, and set the resulting derivative equal to zero: The solution to that equation will give you the value of $x$ where any minimums and/or maximums are going to occur. You'll need to test each solution (if more than one positive solution exists) to see which is a minumum: which gives you the least possible value when $A$ is evaluated there.

Then you'll compute the value of $A$, total area, the value of $x + 20$, and the value of $y + 14$.

Feel free to check back once you've differentiated and found the solutions that solve $A' = 0$ (Hint: after derivating, you may want to multiply through by $x^2$ to simplify $A' = 0$, so it is more easily solvable.

  • $\begingroup$ width of plywood = W+ side margins =W+10 height of plywood =H+top margin+ bottom margin = H+8+6= H+14 Plywood area= (W+10)*(H+14) from equation no.1 W=2000/H So plywood area =(2000/H +10)(H+14) =(2000+10H +28000/H+140) so plywood area = 2140+10H+28000/H So d(plywood area)/dH =0 ------>>>> 0=d(2140+10H+28000/H)/dH So 0=0+10-28000/(H^2) ---->>> H^2=2800 ----->>>> H=52.91 --->>> from equation no.1 W=2000/H=37.796 So height of plywood =H+14 =52.91+14=66.91 ft And width of plywood =W+10 =37.796+10=47.796 ft $\endgroup$ – Michael Rametta Mar 27 '13 at 1:02
  • $\begingroup$ what are the correct answers because i feel this is incorrect $\endgroup$ – Michael Rametta Mar 27 '13 at 1:02
  • $\begingroup$ note that width of plywood = x + 10 + 10 = x + 20. (two side margins: left, right). $x$ is width of printed area, y is height of printed area, so height of plywood is $y + 14$. Plywood area is as stated above $A$. Try it with the correct area formula. $\endgroup$ – Namaste Mar 27 '13 at 1:07
  • $\begingroup$ so whhat is the correct answer $\endgroup$ – Michael Rametta Mar 27 '13 at 2:25
  • $\begingroup$ i got 114 and 40 $\endgroup$ – Michael Rametta Mar 27 '13 at 2:26

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