Q1) Q2) ------------------------------------------------------------------ Q3) A
ID: 2833244 • Letter: Q
Question
Q1)
Q2)
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Q3)
A cup of coffee is made with boiling water (100 degree C) and stands in a room where the temperature is a constant 23degree C. (a) If T(t) is the temperature of the coffee at time t (measured in minutes), what does the differential equation below mean? dT/dt = -k(T - 23) The rate at which the temperature changes is proportional to the difference between the temperature of the coffee and that of the room. The rate at which the temperature changes is proportional to the temperature of the coffee, The rate at which the temperature changes is proportional to the temperature of the room. The rate at which the temperature changes is proportional to the temperature of the coffee and that of the room.(b) What is the sign of k? positive zero negative (c) Solve this differential equation. Your answer will have k in it. Use lower case k. T(t) = (d) If the coffee cools to 90 degree C in 3 minutes, find k. Round your answer to 4 decimal places. (e) How long will it take for the coffee to cool to 60 degree C? Round your answer to one decimal place. minutes The slope field for y' = 0.5(3 + y)(6 - y) is shown in the figure below. Find the equilibrium solutions and state whether they are stable or unstable. (smaller value) y = is a(n) equilibrium. (larger value) y = is a(n) equilibrium. The rate of growth of a tumor is proportional to the size of a tumor. (a) Write a differential equation satisfied by S, the size of the tumor, in mm, as a function of time t. Assume the proportionality constant, k, is positive. dS/dt = (b) Find the general solution to the differential equation. Use C as your constant of integration. S =Explanation / Answer
1.
(a)a
(b)a
(c)23 + 77e^(-kt)
(d)
90 = 23 +77e^(-180k)
=>
e^(180k) = 77/67
=>
k = ln(77/67) / 180 = 0.0008
(e)
60 = 23+77 e^(-kt)
=>
e^(kt) = 77/37
=>
kt = ln(77/37)
=>
t = 15.8 minutes (15 minutes 48 seconds)
2.
-3, unstsable
6, stable
3.
(a)kS,
(b)S = Ce^(kt)
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