Answer to Question #253538 in Calculus for Usman Rashid

Question

In the temperature range between 0C and 7000C the resistance R [in ohms] of a certain platinum resistance thermometer is given by

R = 10 + 0.04124T − 1.779 × 10−5T²

where T is the temperature in degrees Celsius. Where in the interval from 0C to 700C is the resistance of the thermometer most sensitive and least sensitive to temperature changes? [Hint: Consider the size of dR/dT in the

interval 0 ≤ T ≤ 700.].

Answer 1

Sensitivity describes the smallest absolute amount of change that can be detected by a measurement. So, the thermometer is most sensitive at minimal |dR/dT|.

$\frac{dR}{dt}=0.04124-3.558\cdot10^{-5}T$

$\frac{dR}{dt}=0$ at $T=0.04124/3.558\cdot10^{-5}=1.159\cdot10^{-3}\ \degree C$

$\frac{dR}{dt}(0)=0.04124$

$\frac{dR}{dt}(700)=0.04124-3.558\cdot10^{-5}\cdot700=0.16334$

So, the resistance of the thermometer is most sensitive at $T=1.159\cdot10^{-3}\ \degree C$ and is least sensitive at $T=700\degree C$