Q1) ------------------------------------------------------ Q2) Q3) The slope fie
ID: 2833262 • Letter: Q
Question
Q1)
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Q2)
Q3)
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. (a) A refrigerated object is placed in a 68 degree F room. Write a differential equation for H, the temperature of the object at time t. (Use k > 0 for the constant of proportionality.) dH/dt (b) Find the equilibrium solution to the differential equation. H = Determine from the differential equation whether the equilibrium is stable or unstable. stable unstable (c) Give the general solution for the differential equation. H = (d) The temperature of the object is 40 degree F initially, and 48 degree F one hour later. Find the temperature of the object after t = 8 hours. (Round your answer to one decimal place.) degree F A cup of coffee is made with boiling water (100 degree C) and stands in a room where the temperature is a constant 23 degree 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. minutesExplanation / 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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