D I Til and placed it in an oven set to 200° C. After ten minutes she checks the quiche and it has a temperature of T = 20°C. The rate that the temperature of the quiche increa is proportional to the difference in temperature between the oven and the quiche. Write a differential equation to model this scenario and use it to show that T 200 + Ae-kt. Find the value of k in exact form. The quiche is ready when the internal temperature reaches 60° C. How much longe

D I Til and placed it in an oven set to 200° C. After ten minutes she checks the quiche and it has a temperature of T = 20°C. The rate that the temperature of the quiche increa is proportional to the difference in temperature between the oven and the quiche. Write a differential equation to model this scenario and use it to show that T 200 + Ae-kt. Find the value of k in exact form. The quiche is ready when the internal temperature reaches 60° C. How much longe

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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Question

(i)
(ii)
(iii)
Megan has taken a frozen quiche from the freezer with an initial temperature T = 0°C
and placed it in an oven set to 200° C. After ten minutes she checks the quiche and now
it has a temperature of T = 20°C. The rate that the temperature of the quiche increases
is proportional to the difference in temperature between the oven and the quiche.
Write a differential equation to model this scenario and use it to show that
200 + Ae-kt.
T
=
Find the value of k in exact form.
The quiche is ready when the internal temperature reaches 60° C. How much longer
after Megan checked the quiche will it be ready? Answer to the nearest minute.

Expert Solution

Step 1

Let the temperature of the quiche at time 't' be 'T'

Given rate of temperature of the quiche is proportional to difference of temperature of oven and the quiche.

Temperature of the oven is 200⁰C

hence

$\frac{d T}{d t} \propto (T - 200)$

$\frac{d T}{d t} = - k (T - 200)$

where -k is proportionality constant.

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