(1 point) A population P obeys the logistic model. It satisfies the equation dP P(11 - P) for P > 0. dt 1100 (a) The population is increasing when

(c) Assume that P(0) = 3. Find P(74). P(74) =(1 point) Biologists stocked a lake with 400 fish and estimated the carrying capacity (the maximal population for the sh of that species in that lake) to be 9600. The number of fish doubled in the first year. (a) Assuming that the size of the fish population satisfies the logistic equation dP P =kP 1 dt ( K)' determine the constant k, and then solve the equation to find an expression for the size of the population after 12 years. A- = , P(t) : (b) How long will it take for the population to increase to 4800 (half of the carrying capacity)? It will take years. (1 point)Another model for a growth function for a limited population is given by the Gompertz function. which is a solution of the differential equation 0'? K E 6111(3)}? where c is a constant and K is the carrying capacity. (a) Solve this differential equation fore : [1.25, K : 5000: and initial population P0 : 600. Pm = (b) Compute the limiting value of the size of the population. lim PU) = troc- (c) At what value of P does P grow fastest? P : 0313 (1 point) Let Pft) be the performance level of someone learning a skill as a function of the training time t. The derivative represents the rate at which performance improves. If M is the maximum level of performance of which the learner is capable, then a model for learning is given by the differential equation dP E : MM P(t)) where k is a positive constant. Two new workers. David and Andy. were hired for an assembly line. David could process 11 units per minute after one hour and 15 units per minute after two hours. Andy could process 10 units per minute after one hour and 16 units per minute after two hours. Using the above model and assuming that 13(0) 2 0. estimate the maximum number of units per minute that each worker is capable of processing. David: Andy: ('1 point) The tank in the form of a right-circular cone of radius 14 feet and height 13 feet standing on its end. vertex down. is leaking through a circular hole of radius 3 inches. Assume the friction coefficient to be c = 0.6 and g = 32ft/32. Then the equation governing the height h of the leaking water is db. _ dc If the tank is initially full, it will take it seconds to empty. (1 point) Two chemicals A and B are combined to form a chemical C. The rate of the reaction is proportional to the product of the instantaneous amounts of A and B not converted to chemical C. Initially there are 84 grams of A and 90 grams of B, and for each gram of B, 1.4 grams of A is used. It has been observed that 43.5 grams of C is formed in 10 minutes. How much is formed in 30 minutes? What is the limiting amount of C after a long time ? grams of C are formed in 30 minutes grams is the limiting amount of C after a long time