Suppose you have just poured a cup of freshly brewed coffee with temperature 95°C in a room where the temperature is 25°C.Newton's Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings.Therefore, the temperature of the coffee, T(t), satisfies the differential equation=k(T - Troom)where Troom = 25 is the room temperature, and k is some constant.Suppose it is known that the coffee cools at a rate of 2°C per minute when its temperature is 65°C.A. What is the limiting value of the temperature of the coffee?lim T(t) =t-100B. What is the limiting value of the rate of cooling?dTlimt-100 dtdTdtC. Find the constant k in the differential equation.k

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Answered: Suppose you have just poured a cup of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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All temperatures are in degrees Fahrenheit. A cup of coffee at 173 degrees is poured into a mug and left in a room that stays at 68 degrees. After 2 minutes, the coffee is 132 degrees. Assume that the differential equation describing Newton's Law of Cooling is (in this case) dT = k(T - 68) dt What is the temperature of the coffee after 12 minutes? After how many minutes will the coffee be 100 degrees?
Newton's Law of Cooling tells us that the rate of change of the temperature of an object is proportional to the temperature difference between the object and its surroundings. This can be modeled by the differential equation = k(T – A), where T is the temperature of the object after t units of time dt dT have passed, A is the ambient temperature of the object's surroundings, and k is a constant of proportionality. Suppose that a cup of coffee begins at 188 degrees and, after sitting in room temperature of 68 degrees for 18 minutes, the coffee reaches 178 degrees. How long will it take before the coffee reaches 166 degrees? Include at least 2 decimal places in your answer. minutes
The proportion of people in a community infected with a virus that stays active for two weeks is x and the proportion uninfected is y. If the virus spreads only by contact between infected and uninfected members of the community, derive the differential equation that governs the virus spread and solve it assuming the fraction infected at t = 0, is xo. What happens over a long period and what recommendations would you make to contain the spread of the virus
According to Newton’s Law of Cooling, the rate of change of the temperature of an objectcan be modeled using the following differential equation: dT/dt=−κ(T−A) where T is the temperature of the object (in◦F), A is the room temperature (in◦F), and κ is the constant of proportionality. (a) Use an appropriate method to solve the differential equation.Remember that κ and A are constants. (b) Use your answer to (a) to solve the following: On a crime show, a dead body is discovered in a hotel room. A forensic technician recordedthat the victim’s body temperature was 91.4◦F at 6:00 pm. One hour later, the coroner arrived and found that the victim’s body temperature had fallen to 88.7◦F. If the thermostatin the hotel room is set at 68◦F, determine when the victim was murdered.(Assume the victim was a healthy 98.6◦F at the time of death.)
1. Assume that the rate at which a hot body cools is proportional to the difference in temperature between it and its surroundings (Newton's law of cooling). A body is heated to 100°C and placed in air at 10°C. After 1 hour its temperature is 60°C. How much additional time is required for it to cool to 30°C?
A body was found in a room when the room's temperature was 75°F. Let f(t) denote the temperature of the body, t hours from the time of death. According to Newton's law of cooling, f satisfies a differential equation of the form y' = k(T - y). Answer parts (a) - (d) below. (a) Find T. T= 75 ° F (b) After several measurements of the body's temperature, it was determined that when the temperature of the body was 78 degrees, it was decreasing at a rate of 5 degrees per hour. Find k. k = (Round to three decimal places as needed.) (1,1) More
A body was found in a room when the room's temperature was 66°F. Let f(t) denote the temperature of the body, t hours from the time of death. According to Newton's law of cooling, f satisfies a differential equation of the form y' = k(T - y). Answer parts (a) - (d) below. (a) Find T. T= 66 ° F (b) After several measurements of the body's temperature, it was determined that when the temperature of the body was 79 degrees, it was decreasing at a rate of 5 degrees per hour. Find k. k= .385 (Round to three decimal places as needed.) (c) Suppose that at the time of death, the body's temperature was about normal, say 98°F. Determine f(t). -.385t f(t) = 32 e + 66 (Use integers or decimals for any numbers in the expression.) (d) When the body was discovered, its temperature was 86°F. Determine how long ago the person died. The person died hours before discovery. (Do not round until the final answer. Then round to three decimal places as needed.)
Newton’s law of cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature (i.e., the temperature of its surroundings). We let (t) denote time, T(t) be the temperature of the object at time (t), (T0 = T(0)) (the object’s temperature at (t = 0)), and Tomg be the constant ambient temperature. Set u(t) = T(t) - Tomg and show from Newton’s law of cooling that (u) satisfies the initial value problem (*) u’(t) = ku(t), u(0) = T0 - Tomg for a constant (k). Solve the initial value problem (*), i.e., find u(t)) (expressed in terms of (k), T0, and Tomg . Find T(t).
Newton's Law of Cooling tells us that the rate of change of the temperature of an object is proportional to the temperature difference between the object and its surroundings. This can be modeled by the differential equation =k(TA), where T is the temperature of the object after t units of time have passed, A is dT dt the ambient temperature of the object's surroundings, and k is a constant of proportionality. Suppose that a cup of coffee begins at 179 degrees and, after sitting in room temperature of 62 degrees for 14 minutes, the coffee reaches 170 degrees. How long will it take before the coffee reaches 160 degrees? Include at least 2 decimal places in your answer. minutes
Newton's Law of Cooling states that the rate at which the temperature of a cooling object decreases and the rate at which a warming object increases are proportional to the difference between the temperature of the object and the temperature of the surrounding medium. The differential equation formula is given as: dT = k (T – A) dt - T = temperature of object at time t А ambient temperature (aka temperature of the surrounding medium) k = constant of proportionality Okay, now here's the problem. A glass of delicious orange soda (with an initial temperature of 34°F) is placed in a room with a constant temperature of 72°F. One hour later, the temperature of the orange soda is 52°F. Find the exact simplified value of k. Hint: solve the differential equation for T and note that A is a constant.
A virus has already infected 4 million people in a country before a vaccine becomes available to the public. Once vaccination starts, the number of infected people decreases at a rate that is proportional to the cube of the number of infected people. One month later, there are still 2 million people that are infected. Let p be the number of infected people (in millions) and t be the number of months since vaccination has started. (a) Find a differential equation and initial conditions to model the situation. (b) Solve the differential equation to find pp as a function of t (c) How many people are still infected three months after vaccination starts?
A 288 pound object is suspended from a spring. The spring stretches an extra 9 inches with the weight attached. The spring-and-mass system is then submerged in a viscous fluid that exerts 16 pounds of force when the mass has velocity 2 ft/sec. What is the differential equation for this spring-and-mass system? (Use u as the dependent variable.) This system is ? For what value of y would the system be critically damped? (if the given system is already critically damped, enter the given value for y.)

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