The rate at which a body cools also depends on its exposed surface area S. If S is a constant, then a modification of (1) is dT kS(T – Tm) dt where k >0 and Tm is a constant. Suppose that two cups A and B are filled with coffee at the same time. Initially, the temperature of the coffee is 120° F. The exposed surface area of the coffee in cup B is twice the surface area of the coffee in cup A. After 30 min the temperature of the coffee in cup A is 90° F. If Tm =70° F, then what is the temperature of the coffee in cup B after 30 min?

The rate at which a body cools also depends on its exposed surface area S. If S is a constant, then a modification of (1) is dT kS(T – Tm) dt where k >0 and Tm is a constant. Suppose that two cups A and B are filled with coffee at the same time. Initially, the temperature of the coffee is 120° F. The exposed surface area of the coffee in cup B is twice the surface area of the coffee in cup A. After 30 min the temperature of the coffee in cup A is 90° F. If Tm =70° F, then what is the temperature of the coffee in cup B after 30 min?

Name: Q1. The mathematical formulation of Newton's empirical law of cooling
Uploaded: 2024-03-20T06:46:42.000Z
Channel: Bartleby
Description: Q1. The mathematical formulation of Newton's empirical law of cooling/warming of an object isgiven by the linear first-order differential equationdT= k(T – Tm)(1)dtwhere k is a constant of proportionality, T(t) is the temperature of the object for t

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Newton's law of cooling says that a hot object cools rapidly when the difference between its…

Q: When a vertical beam of light passes through a transparent medium, the reason that its intensity I…

A: We have to find What is the beam intensity at 15 feet below the surface.

Q: Skydiving If a body of mass m falling from rest under the action of gravity encounters an air…

A: Given that a body of mass m falling from rest under the action of gravity.The differential equation…

Q: A certain rumor spreads through a community at the rate dy/dt=3y(1+y) , where y is the portion of…

Q: Newton's law of cooling states that the rate of change in the temperature T(t) of a body is…

A: Given Data: Newton's law of cooling states that the rate of change in the temperature T(t) of a…

Q: The kinetic energy E of a rocket of mass m, traveling at velocity v is given by the formula E = mv²…

A: Now formula for kinetic energy and the value of mass are given.

Q: A tank contains 210 liters of fluid in which 20 grams of salt is dissolved. Pure water is then…

Q: Suppose the velocity of an object at time t is given by v(t) + m/s. 4(21+5)² 100 Find the average…

Q: A balloon leaks air (changes in volume) at a rate of one-third the volume per minute. Write an…

A: Given : A balloon leaks air (changes in volume) at a rate of one-third the volume per minute.

Q: A tank contains 14,000 L of brine with 21 kg of dissolved salt. Pure water enters the tank at a rate…

Q: A beaker of liquid was placed in a fridge. The rate of cooling is given by dT — k(Т — Тр), k> 0 dt…

Q: The power P (in W) in an electric circuit is the product of the voltage V and the current I (in A).…

Q: A tank contains 200 liters of fluid in which 50 grams of salt is dissolved. Pure water is then…

Q: A piece of chicken is boiled until its internal temperature reaches 165 °F. The chicken is then…

A: Given the formula for Newton's cooling law…

Q: A tank contains 210 liters of fluid in which 50 grams of salt is dissolved. Pure water is then…

A: To find The rate at which the number of grams of salt in the tank is changing at time t.

Q: A metal bar of initial temperature 15 degrees C is placed in a a vat of boiling water (100 degrees…

Q: Newton's law of cooling says that the rate at which a body cools is proportional to the difference…

Q: Newton's law of cooling states that the rate of change in the temperature T(t) of a body is…

A: please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Topic Video

Question

Q1. The mathematical formulation of Newton's empirical law of cooling/warming of an object is
given by the linear first-order differential equation
dT
= k(T – Tm)
(1)
dt
where k is a constant of proportionality, T(t) is the temperature of the object for t> 0, and Tm is
the ambient temperature that is, the temperature of the medium around the object.
The rate at which a body cools also depends on its exposed surface area S. If S is a constant, then
a modification of (1) is
dT
kS(T – Tm)
(2)
dt
where k> 0 and Tm is a constant. Suppose that two cups A and B are filled with coffee at the
same time. Initially, the temperature of the coffee is 120° F. The exposed surface area of the
coffee in cup B is twice the surface area of the coffee in cup A. After 30 min the temperature of
the coffee in cup A is 90° F. If Tm =70° F, then what is the temperature of the coffee in cup B
after 30 min?

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps with 8 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Chain Rule$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

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).
A tank contains 210 liters of fluid in which 50 grams of salt is dissolved. Pure water is then pumped into the tank at a rate of 3 L/min; the well-mixed solution is pumped out at the same rate. Let the A(t) be the number of grams of salt in the tank at time t. Find the rate at which the number of grams of salt in the tank is changing at time t. Find the number A(t) of grams of salt in the tank at time t.
At what speed should a fish swim upstream so as to reach its destination with the least expenditure of energy? The energy depends on the friction of the fish through the water and on the duration of the trip. If the fish swims with velocity v, the energy has been found experimentally to be proportional to (for constant k > 2) times the duration of the trip. A distance of s miles against a current of speed c requires time -c (distance divided by speed). The energy required is then proportional to k = 3, minimizing energy is equivalent to minimizing |- v³ E(v) = = V C vk s v-c. For Find the speed v with which the fish should swim in order to minimize its expenditure E(v). (Your answer will depend on c, the speed of the current.)
Automobile traffic passes a point P on a road of width w feet with an average rate of R vehicles per second. Although the arrival of automobiles is irregular, traffic engineers have found that the average waiting time T until there is a gap in traffic of at least 1 seconds is approximately T = te Riseconds. A pedestrian walking at a speed of 3.3 ft/s requires t =s to cross the road. Therefore, the average time the pedestrian will have to wait before crossing is f(w, R) = () WR/3.3 What is the pedestrian's average waiting time if w = 24 ft and R = 0.2 vehicle per second? (Use decimal notation. Give your answer to two decimal places.) (= Use the Linear Approximation to estimate the increase in waiting time if w is increased to 26 ft. (Use decimal notation. Give your answer to two decimal places.) Af = 31.15 t = Estimate the waiting time if the width is increased to 26 ft and R decreases to 0.18. (Use decimal notation. Give your answer to two decimal places.) 6.37 A = 32.48 Incorrect What…
Water flows into a tank, and a certain part of it drains out through a valve. The volume v (in cubic feet) of water in the tank at time t satisfies the equation dv/dt = 5 − (v/3). If the process continues for a long time, how much water will be in the tank?a. 13 cubic feetb. 14 cubic feetc. 15 cubic feetd. 16 cubic feet