Expert: Scotto - 3/15/2010

Question
Hi, I have 2 calculus problems regarding deravatives and max min (optimization). Any help is appreciated.

1. While in refrigerated storage, 1000 m^3 of apples spoil at the rate of 20 m^3/month. The September price is $2.5/m^3 and the price increases at $1.25 m^3/month. Storage prices are $750/month. If the apples are stored in september, when should they be sold for a maximum gross profit?

2. A ship is 25 nautical miles due north of a drifting derelict vessel. if the sailing ship sails south at 4 knots (nautical miles per hour) while the derelict drifts east at 3 knots, find the distance of closest approach of two ships?

Answer
1. The number of apples is N = (1000-20M) where M=months.
The price is P = 2.50 + 1.25M.

The amount made is C = NP.  Compute C, multiply it out, take the derivative with respect to M,
set to 0, sovlve for M, and this determines how many months to wait to sell them.


2. Assuming a knot is a nautical mile/hour, the ship up north is N = 25 - 4H
where N is the distance to the ship north and H is hours.

The other ship is called E and its position is E = 3H.

The distance D squared is D² = N² + E².

Put in what N and E are, and they are variables of H.
Take the derivative as is.  This is because when D² is at a maximum, and D is positive,
D is also at a maximum.

Set it equal to 0 and solve for H.

Put H back into to N and E to find the position of the two ships.
The answer for N will be how far north the 1st ship is and
the answer for E will be how far east the 2nd ship is.