A digital medical thermometer contains a thermistor that has a temperature coefficient of resistance of negative 0.060 inverse degrees Celsius when it is at the same temperature as a patient. What is the patient’s temperature if the thermistor’s resistance at that temperature is 82 percent of its value at 37 degrees Celsius?
All right, in this example, we have a thermistor, which is a device used to measure temperature based on the increase or decrease of the resistance of a material. Our thermistor is being used as a medical thermometer, so we imagine the doctor may leave it in a patient’s mouth until the temperature of the thermometer is the same as the patient.
We want to solve for the patient’s temperature based on the fact that the thermistor’s resistance at that temperature, the patient’s temperature, is 82 percent of its previous value at 37 degrees Celsius. We’re talking then about a change in temperature from 37 degrees Celsius corresponding to a change in resistance. These two changes are related to one another in a mathematical relationship we can recall.
We can say that, for our resistor that changes temperature, the change in resistance that results is equal to a baseline or reference resistance value multiplied by the temperature coefficient of resistance, 𝛼, all times the change in temperature, Δ𝑇, of the resistor. And in our scenario, we’re told a bit about what Δ𝑅 is.
We know that if the baseline resistance value of a resistor was taken at a temperature of 37 degrees Celsius and if we call that baseline resistance 𝑅 sub zero, then the change in resistance, Δ𝑅, reflects the fact that 𝑅 sub zero has decreased to 82 percent of its original value. In other words, the resistor has lost 18 percent of the resistance it had at first. If we write this percentage as a decimal, then we can use that decimal value in front of 𝑅 sub zero. And it’s this product, negative 0.18 times 𝑅 sub zero, which is equal in our particular instance to Δ𝑅.
Again, this expression here means that our original baseline resistance has decreased — that’s what the minus sign is about — by 18 percent, represented by the decimal 0.18, so far so good in solving for this expression for Δ𝑅.
Now let’s continue writing the rest of this equation on the right-hand side. We can say that negative 0.18𝑅 sub zero is equal to 𝑅 sub zero times 𝛼, the temperature coefficient of resistance, times Δ𝑇, the change in temperature of the thermometer.
We see first off in this equation that 𝑅 sub zero, our reference resistance, cancels out. We can then plug in for our value of 𝛼, which is negative 0.060 inverse degrees Celsius. The negative sign in 𝛼 is interesting. It means that as our temperature drops, our resistance actually goes up.
Next, to isolate Δ𝑇 on one side of this equation, we’ll divide both sides by 𝛼. When we do, we see two interesting things. First of all, the minus signs in numerator and denominator cancel out. And second, the overall units of this fraction will be degrees Celsius. When we calculate this fraction, we find a result of three degrees Celsius, but that’s not our final answer.
Remember that what we’ve solved for is the change in temperature, Δ𝑇, which started out at 37 degrees Celsius. That means that the patient’s temperature, which we can call 𝑇 sub 𝑃, is equal to 37 degrees Celsius plus Δ𝑇. 37 degrees Celsius plus three degrees Celsius is 40 degrees Celsius. That’s the temperature of the patient as measured by this thermometer.