A poster is to contain 108cm^2 of printed matter, with margins of 6cm at the top and bottom and 2cm on the sides. What is the minimum cost of the poster if it is to be made of material costing 20cents/cm?

NOTE: Draw a rectangle figure and let x denote the horizontal dimension of the printed matter and y the vertical dimension. The find a formula involving x and y for the area of the poster. You need to minimize the area. Then write a formula involving x and y for the cost C of material for the poster.

Question

A poster is to contain 108cm^2 of printed matter, with margins of 6cm at the top and bottom and 2cm on the sides. What is the minimum cost of the poster if it is to be made of material costing 20cents/cm?

NOTE: Draw a rectangle figure and let x denote the horizontal dimension of the printed matter and y  the vertical dimension. The find a formula involving x and y for the area of the poster. You need to minimize the area. Then write a formula involving x and y for the cost C of material for the poster.

anonymous · Answer

I'v worked this problem several times and cant seem to come to the right answer. I need some serious help

campbell_st · Answer

the area of the printed material is 108 cm^2

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this means xy = 108cm^2     - the printed area

you will need  y = 108/x      for later

then the width of the entire poster is x + 2+2  or   (x + 4)    2cm on either side
the length of the entire poster is       y + 6 + 6  or (y + 12)    6cm on to and bottom

then the area is 

A = (x+4)(y+12) 

A= xy + 4y + 12x + 48

since xy = 108 then 

A = 108 + 4y + 12x + 48

A = 4y + 12x + 156

using y = 108/x

A = 4(108/x) + 12x + 156

A = 432/x + 12x + 156

I'm unclear of the cost as to whether it is 20cents/cm or 20cents/cm^2

cost = Area x 0.2 

Cost = 0.2(432/x +12x + 156)

you'll need to differentiate dC/dx

then let dC/dx = 0 and solve 

to find the values of x to give the minimum cost