Profit = revenue-cost

Revenue = pq (price times quantity sold)

Revenue = (1700-7q)(q)

R = 1700q-7q^2

Profit = (1700q-7q^2) - (16000+500q-1.6q^2+0.004q^3)

P = 1700q-7q^2-16000-500q+1.6q^2-0.004q^3

dP/dq = 1700-14q-500+3.2q-0.012q^2

set dP/dq=0

1700-14q-500+3.2q-0.012q^2 =0

-0.012q^2-10.8q+1200 = 0

0.012q^2+10.8q-1200 =0

multiply by 1000

12q^2+10800q-1200000 = 0

This equation is of form ax^2+bx+c=0

a = 12 b = 10800 c = -1200000

x=[-b+/-sqrt(b^2-4ac)]/2a]

x=[-10800 +/-sqrt(10800^2-4(12)(-1200000)]/(2)(12)

discriminant is b^2-4ac =174240000

x=[-10800 +√(174240000)] / (2)(12)

x=[-10800 -√(174240000)] / (2)(12)

x=[-10800+13200] / 24

x=[-10800-13200] / 24

The roots are 100 and -1000

q=-1000 or q=100

q=100 (quantity that must be produced to make the production level maximum)

d^2P/dq^2 = -0.024q-10.8

when q=100, d^2P/dq^2 = -13.2 < 0 so production is maximum when q=100