Could someone please help me figure this problem out?
A company, producing muesli bars, faces the demand equation x3+xp+p3=143 between price and quantity, where the quantity x is measured in millions. The cost to produce 1 million units increase over time according to the function k(t)=200t. We are presently in time t=25. The company has achieved a constant increase of the profit of 1000000 dollars per year and expects to maintain this standard.
a) Presently, 5000000 units are produced and sold at a price of 2dollars per piece. At what rate is the production level x increasing or decreasing? What about the rate of change for the price?
b) More generally, try to express the rate of change of x (over time) in terms of x and p. Can you express it in terms of x alone? Can you express it in terms of time t? If not, try to describe why not.
c) Can you explain the difference of the setting of this exercise and the ordinary related rates situation?