Question

A thermometer measuring body temperature follows a first-order response with a time constant of 40 seconds. The instrument will reach 95% of its steady-state output at __ seconds.

• A. 60
• B. 80
• C. 120
• D. 160

Hint:

In first-order systems, the time to reach a certain percentage of steady-state output is related to the time constant and can be calculated using the formula for exponential decay.

Answer 1

The Correct Option is C

Step 1: Understand the first-order response equation.

For a first-order system, the time required to reach a certain percentage of the steady-state value can be determined using the formula:
\[ \text{Percentage of steady-state} = 100 \left(1 - e^{-\frac{t}{\tau}}\right), \]
where \( \tau \) is the time constant, and \( t \) is the time.

Step 2: Calculate the time to reach 95% steady-state.

We want to find the time \( t \) at which the instrument reaches 95% of its steady-state output. Setting the percentage equal to 95%, we have:
\[ 95 = 100 \left( 1 - e^{-\frac{t}{40}} \right) \]
Simplifying:
\[ 0.95 = 1 - e^{-\frac{t}{40}}, \quad e^{-\frac{t}{40}} = 0.05 \]
Taking the natural logarithm of both sides:
\[ -\frac{t}{40} = \ln(0.05), \quad t = -40 \times \ln(0.05) \]
Using a calculator:
\[ t \approx 120 \, \text{seconds} \]

Final Answer:
The instrument will reach 95% of its steady-state output in \( \boxed{120} \) seconds.