A squash 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