A printer needs to make a poster that will have a total of 500 cm2 that will have 3 cm margins on the sides and 2 cm margins on the top and bottom. What dimensions of the poser will give the largest printed area?

The largest printed area is 381.43 cm^2.

Step-by-step explanation:

The width of the full poster = 500/x

Length of the printed area = l = x - 4

Width of the printed area = w = (500/x) - 2

Area of the printed space = (x - 4) * ((500/x) - 2)

Now, take derivative of the area

A' = (x - 4) * (-500/x^2) + ((500/x) - 2)

A' = (2000/x^2) - 2

A' = (2000-2x^2) / x^2

put derivative equal to zero like A' = 0

(2000-2x^2) / x^2 = 0

2000 = 2x^2

x^2 = 1000

x = 31.62

So, the length of the original poster is = x = 31.62 cm

The width of the full poster = 500/x = 15.81 cm

l = x - 4 = 27.62

w = 500/x - 2 = 13.81

Therefore, The area of space available for printing is = l * w

= 27.62 * 13.81 = 381.43 cm^2.