A small metal sphere at temperature (in degrees Celsius) cools in a room whose…

MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations

Problem

A small metal sphere at temperature

T

(in degrees Celsius) cools in a room whose ambient temperature is

20^{\circ}\text{C}

. Its temperature change is modeled by

\frac{dT}{dt}=-k\,(T-20), \qquad k>0,

where

k

is a material–dependent cooling constant. A student wishes to compare the slope fields for

k=0.1

and

k=0.4

(time measured in minutes). (a) For each value of

k

, indicate the

T

–value(s) where the short line segments in the slope field are horizontal. Justify your answer algebraically. (b) On separate axes of identical scale, sketch qualitative slope fields for

k=0.1

and

k=0.4

. Clearly label the parameter value on each sketch. (c) Describe two visual differences between the two slope fields that would allow someone to determine which sketch corresponds to the larger value of

k

without performing any calculations. (d) A sphere is removed from an oven at

120^{\circ}\text{C}

and allowed to cool according to the model with

k=0.4

. Using only your sketch from part (b), estimate whether the temperature will drop below

40^{\circ}\text{C}

within the first 10 minutes and explain your reasoning.

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