h = 180 cm
w = 108 cm
First, let's express width (W) as a function of height (H).
19440 = WH
19440 / H = W
Now let's express printed area using W and H
A = (W - 2*6)(H - 2*10)
A = (W - 12)(H - 20)
Now substitute the equation for W into the formula
A = (W - 12)(H - 20)
A = (19440/H - 12)(H - 20)
And expand and simplify
A = (19440/H - 12)(H - 20)
A = H*19440/H - 12H - 20*19440/H + 240
A = 19440 - 12H - 388800/H + 240
A = 19680 - 12H - 388800/H
Since we're looking for a maximum, that will happen when the slope of the function is 0. And the first derivative of the function will calculate the slope of each point of the function. So let's calculate the first derivative.
A = 19680 - 12H - 388800/H
A = 19680*H^0 - 12H - 388800*H^(-1)
A' = 0*19680*H^(0-1) - 1*12H^(1-1) - (-1)*388800*H^(-1-1)
A' = 0*19680*H^(-1) - 1*12H^(0) - (-1)*388800*H^(-2)
A' = 0*19680/H - 1*12*1 - (-1)*388800/H^2
A' = 0 - 12 + 388800/H^2
A' = 388800/H^2 - 12
Now set to 0 and solve for H
A' = 388800/H^2 - 12
0 = 388800/H^2 - 12
12 = 388800/H^2
12H^2 = 388800
H^2 = 388800/12 = 32400
H = 180
So the optimal height is 180 cm. The width will be 19440 / 180 = 108 cm
