Newton's law of cooling Indicates that the temperature of a warm object, such as a cake coming out of the oven, will decrease exponentially with time and will approach the temperature of the surrounding air. The temperature T t is modeled by T t = T a + T 0 − T a e − k t . In this model. T a represents the temperature of the surrounding air, T 0 represents the initial temperature of the object, and t is the dine after the object rearm cooling. The value of k is a constant of proportion relating the temperature of the object to its rate of temperature change. Use this model for Exercises 59-60. Water in a water heater is originally 122°F . The water heater is shut off and the water cools to the temperature of the surrounding air, which is 60°F . The water cools slowly because of the insulation inside the heater, and the value of k is measured as 0.00351. a. Write a function that models the temperature T t in o F of the water t hours after the water heater is Shut off. b. what is the temperature of the water 12 hr after the heater is shut off? Round to the nearest degree. c. Dominic does not like to shower with water less than 115°F . If Dominic mho 24 hr, will the water still be warm enough for a shower?
Newton's law of cooling Indicates that the temperature of a warm object, such as a cake coming out of the oven, will decrease exponentially with time and will approach the temperature of the surrounding air. The temperature T t is modeled by T t = T a + T 0 − T a e − k t . In this model. T a represents the temperature of the surrounding air, T 0 represents the initial temperature of the object, and t is the dine after the object rearm cooling. The value of k is a constant of proportion relating the temperature of the object to its rate of temperature change. Use this model for Exercises 59-60. Water in a water heater is originally 122°F . The water heater is shut off and the water cools to the temperature of the surrounding air, which is 60°F . The water cools slowly because of the insulation inside the heater, and the value of k is measured as 0.00351. a. Write a function that models the temperature T t in o F of the water t hours after the water heater is Shut off. b. what is the temperature of the water 12 hr after the heater is shut off? Round to the nearest degree. c. Dominic does not like to shower with water less than 115°F . If Dominic mho 24 hr, will the water still be warm enough for a shower?
Solution Summary: The author explains Newton's law of cooling, wherein the temperature function is T(t)=60+62e-0.00351t
Newton's law of cooling Indicates that the temperature of a warm object, such as a cake coming out of the oven, will decrease exponentially with time and will approach the temperature of the surrounding air. The temperature
T
t
is modeled by
T
t
=
T
a
+
T
0
−
T
a
e
−
k
t
.
In this model.
T
a
represents the temperature of the surrounding air,
T
0
represents the initial temperature of the object, and t is the dine after the object rearm cooling. The value of k is a constant of proportion relating the temperature of the object to its rate of temperature change. Use this model for Exercises 59-60.
Water in a water heater is originally
122°F
. The water heater is shut off and the water cools to the temperature of the surrounding air, which is
60°F
. The water cools slowly because of the insulation inside the heater, and the value of k is measured as 0.00351.
a. Write a function that models the temperature
T
t
in
o
F
of the water t hours after the water heater is Shut off.
b. what is the temperature of the water 12 hr after the heater is shut off? Round to the nearest degree.
c. Dominic does not like to shower with water less than
115°F
. If Dominic mho 24 hr, will the water still be warm enough for a shower?
2. Consider the following:
Prove that x, x2, and 1/x are the solutions to the homogeneous equation
corresponding to x³y"" + x²y" + 2xy' + 2y = 2x4.
b. use variation of parameters to find a particular solution and complete the general
solution to the differential equation. I am interested in process. You may use a
computer for integration, finding determinants and doing Kramer's.
3. A spring is stretched 6 in. by a mass that weighs 8 lb. The mass is attached to a dashpot
mechanism that has a damping constant of 0.25 lb-sec./ft. and is acted on by an external
force of 4 cos 2t lb.
a. Set-up the differential equation and initial value problem for the system.
b. Write the function in phase-amplitude form.
C.
Determine the transient solution to the system. Show your work.
d. Determine the steady state of this system. Show your work.
e.
Is the system underdamped, overdamped or critically damped? Explain what this
means for the system.
4. Suppose that you have a circuit with a resistance of 20, inductance of 14 H and a
capacitance of 11 F. An EMF with equation of E(t) = 6 cos 4t supplies a continuous charge
60
to the circuit. Suppose that the q(0)= 8 V and the q'(0)=7. Use this information to answer the
following questions
a. Find the function that models the charge of this circuit.
b. Is the circuit underdamped, overdamped or critically damped?
College Algebra with Modeling & Visualization (5th Edition)
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