SOLUTION: A rectangular cardboard poster is to have a 96 - square - inch rectangular section of printed material, a 2 - inch border top and bottom, and a 3 - inch border on each side

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Question 1155489: A rectangular cardboard poster is to have a 96 - square - inch rectangular section of printed material, a 2 - inch border top and bottom, and a 3 - inch border on each side. Find the dimensions and area of the smallest poster that meets these specifications. (Note; Let x and y be the dimensions of the 96 - square - inch area.)
Answer by ikleyn(52786) About Me  (Show Source):
You can put this solution on YOUR website!
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They want you to minimize

    f(x,y) = (x-6)*(y-4)  under the condition  x*y = 96.  (1)

where x and y are the dimensions of the poster.


In other words, from (1), you need to minimize

    f(x,y) = xy - 4x - 6y + 24 = 96 - 4x - 6y + 24 = -4x - 6y + 120.


From (1), you have  y = 96%2Fx, so the function f(x,y) takes the form

    g(x) = -4x - 6%2A%2896%2Fx%29 + 120 = -4x - 576%2Fx + 120.    (2)


So, you differentiate (2), and you get

    g'(x) = -4 + 576%2Fx%5E2.


Equate it to zero

    g'(x) = 0 = -4 + 576%2Fx%5E2.


So, to find x, you have this equation

    4x%5E2 = 576,

    x%5E2 = 576%2F4 = 144

     x = sqrt%28144%29 = 12 centimeters.


Then y = 96%2F12 = 8 cm.


Thus the dimension of the poster are  12 inches width and 8 inches height.

Solved.