Answer:
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 .
