A solid metal sphere cools as it is removed from a furnace

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

A solid metal sphere cools as it is removed from a furnace. The temperature

T

of the metal (in degrees Celsius) is a differentiable function of its radius

r

(in centimeters), and the radius is itself a differentiable function of time

t

(in minutes) as the sphere contracts while cooling. Suppose

T=T(r)

and

r=r(t)

. Answer the following questions about the way changes in radius and temperature are related over time. (a) Describe in words how an infinitesimal change in time produces an infinitesimal change in temperature for the sphere. In your answer, identify the intermediate quantity and explain the logical chain connecting

t

T

. State the units of

\dfrac{dT}{dt}

\dfrac{dT}{dr}

, and

\dfrac{dr}{dt}

. No calculations are required. (b) At a certain instant the sphere has radius

6\text{ cm}

and is shrinking so that

\dfrac{dr}{dt}=-0.04\text{ cm}/\text{min}

. The rate of change of temperature with respect to radius at that radius is

\dfrac{dT}{dr}=-15\text{ °C}/\text{cm}

. Without computing

\dfrac{dT}{dt}

explicitly, determine the sign (positive, negative, or zero) of

\dfrac{dT}{dt}

at that instant and explain your reasoning in complete sentences. (c) Describe a physical situation, consistent with the information above, in which

\dfrac{dT}{dt}=0

even though

\dfrac{dr}{dt}\neq 0

. Explain what must be true about

\dfrac{dT}{dr}

in that circumstance.

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