Given: P(x) = -0.4x^{2} + fx - m

design fee (f) = $80

monthly studio rent (m) = $1,600

# of award designs sold = x

==> P(x) = -0.4x^{2} + 80x - 1600

Notice that this is a quadratic equation (standard form: P = ax^{2} + bx + c), which is parabolic. Parabolic functions open upwards when the leading coefficient ('a') is positive and open downwards when 'a' is negative. Since the leading coefficient in this equation (-0.4) is negative, the parabola opens downwards which means that the x-coordinate of the its vertex is the maximum value of the function and is given by: x = -b/(2a).

The profit when 50 award designs are sold in a month is determined as follows:

x = 50

P(50) = -0.4(50)^{2} + 80(50) - 1600

= -0.4(2500) + 4000 - 1600

= -1000 + 4000 - 1600

= 1400

Therefore, the profit (P) when 50 award designs are sold is $1,400.

To maximize profit, the maximum value of x (award designs) must be sold. As mentioned above, the max value is given by the x-coordinte of the vertex of the parabola: x = -b/(2a)

from the equation ==> b = 80 , a = -0.4

x = -80/(2·-0.4)

= -80/-0.8

= 100

Thus, in order to maximize profit 100 award designs must be sold.

That maximum profit is given by P(x) when x = 100:

P(100) = -0.4(100)^{2} + 80(100) - 1600

= -0.4(10000) + 8000 - 1600

= -4000 + 8000 - 1600

= 2,400

Thus, the maximum profit is $2,400.