A freshly baked loaf is removed from an oven at and placed on a cooling rack in a…

PHYSICS · Multi-part · CBSE · Physics · Thermal Properties of Matter · Thermal Properties of Matter

Problem

A freshly baked loaf is removed from an oven at

200^{\circ}\text{C}

and placed on a cooling rack in a kitchen whose air temperature is

25^{\circ}\text{C}

. After 15 minutes, the loaf is moved to a porch where the air temperature is a constant

10^{\circ}\text{C}

. Let

T(t)

denote the temperature (in

^{\circ}\text{C}

) of the loaf, where

t

is measured in minutes from the instant it leaves the oven. (a) For

0\le t\le 15

, write a differential equation satisfied by

T(t)

that incorporates the kitchen air temperature. (b) Solve the differential equation from part (a) to obtain an explicit expression for

T(t)

on the interval

0\le t\le 15

, given that

T(0)=200

and

T(15)=140

. (c) Using the temperature you found at

t=15

, write a differential equation for

T(t)

on the interval

t\ge 15

that reflects the new porch air temperature. (d) Solve the differential equation in part (c) to obtain an explicit expression for

T(t)

t\ge 15

. (e) Determine the total time, measured from when the loaf leaves the oven, required for the loaf to cool to

40^{\circ}\text{C}

. Round your answer to the nearest minute.