A drug is carried into an organ of volume cm³ by a liquid that enters the organ at the rate of p cm³/sec and leaves it at the rate of q cm³/sec. The concentration of the drug in the liquid entering the organ is r g/cm³. If the concentration of the drug in the organ at time is increasing at the rate of g/cm³/sec, and the concentration of the drug in the organ initially is g/cm³, find the concentration of the drug in the organ at time .

Question

A drug is carried into an organ of volume

cm³ by a liquid that enters the organ at the rate of p cm³/sec and leaves it at the rate of q cm³/sec. The concentration of the drug in the liquid entering the organ is r g/cm³. If the concentration of the drug in the organ at time

is increasing at the rate of

g/cm³/sec, and the concentration of the drug in the organ initially is

g/cm³, find the concentration of the drug in the organ at time

.

Quizplus · Accepted Answer

Let V(t) be the volume of the liquid in the organ at time t, and C(t) be the concentration of the drug in the organ at time t. We are given: - V(0) = 11 cm^3 - dC/dt = k for some constant k - C(0) = r We want to find C(T), where T is the time we are interested in. From the given information, we can set up a differential equation for V(t): dV/dt = p - q Solving this differential equation gives: V(t) = V(0) + (p-q)t Substituting in the given value for V(0) and simplifying, we get: V(t) = 11 + (p-q)t Next, we can set up a differential equation for C(t): dC/dt = k This differential equation has the solution: C(t) = C(0) + kt Substituting in the given value for C(0) and simplifying, we get: C(t) = r + kt To find k, we can use the given information that the concentration of the drug in the organ is increasing at a rate of 11eb651f_cb26_0496_b0a1_2ab5eee51540_TB8255_11 g/cm³/sec. This means: dC/dt = 11eb651f_cb26_0497_b0a1_2ab5eee51540_TB8255_11 Substituting this into the differential equation for C(t), we get: 11eb651f_cb26_0498_b0a1_2ab5eee51540_TB8255_11 = k Therefore, the solution for C(t) is: C(t) = r + 11eb651f_cb26_0499_b0a1_2ab5eee51540_TB8255_11t Finally, to find C(T) we can substitute T = 11eb651f_cb26_0500_b0a1_90d18eb0daed_TB8255_11 into the solution for C(t) and simplify: C(T) = r + 11eb651f_cb26_0501_b0a1_2ab5eee51540_TB8255_11T C(T) = r + 11eb651f_cb26_0501_b0a1_2ab5eee51540_TB8255_11(11 + (p-q)T) C(T) = r + 11eb651f_cb26_0502_b0a1_2ab5eee51540_TB8255_11(11) + 11eb651f_cb26_0503_b0a1_2ab5eee51540_TB8255_11(p-q)T Substituting in the given values for r, p, q, and the rate of increase, we get: C(T) = 11eb651f_cb26_0504_b0a1_952d6a34b0a2_TB8255_11 Therefore, the answer is C.

A Drug Is Carried into an Organ of Volume