Asquash patch currently has 120 pounds of butternut squash. for each additional day in the patch, the amount of butternut squash increases by 6 pounds. if the price of butternut squash is currently 80 cents per pound, but decreases by 2 cents each day, how long should the butternut squash grow to maximize the profit?

Okay let's say that our supply is:

s(t)=120+6t

Now let's say that the price is:

p(t)=0.8-0.02t

Finally we can say that revenue is:  (price times supply)

r(t)=[p(t)]*[s(t)]

r(t)=(0.8-0.02t)(120+6t)

r(t)=96+4.8t-2.4t-0.12t^2

r(t)=-0.12t^2+2.4t+96  now that we have the revenue function we take the derivatives...

dr/dt=-0.24t+2.4, d2r/dt2=-0.24

Since the acceleration is a constant negative we can say that when the velocity, dr/dt=0, it is the absolute maximum for r(t)

dr/dt=0 only when 0.24t=2.4,  t=10 days

So growing the squash for 10 days will maximize profit.  (technically we just used revenue because we had no information as to the cost of growing :P)

If you'd care to check you could simply evaluate r(t) at t=10 and 10±d

r(t)=-0.12t^2+2.4t+96

r(10)=$108,  

r(9.99)=$107.999988

r(10.01)=$107.999988

However, the changes in squash and price take place in daily increments and not hundredths of a day, the differences from the maximum profit will be much larger than shown above

r(9)=$107.88, r(11)=$107.88