Q.42 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.
(Round off to the nearest integer)
(A) 60
(B) 80
(C) 120
(D) 160
95% of their steady-state value after about
three time constants.
For a time constant of 40 seconds, the required time is
120 seconds.
Correct Answer: (C) 120 seconds
Correct Answer Explanation
A first-order system follows the step response equation:
y(t) = yss (1 − e−t/τ)
where:
- τ = time constant
- yss = steady-state value
To find the time required to reach 95% of the steady-state value:
0.95 = 1 − e−t/τ
Rearranging:
e−t/τ = 0.05
Taking the natural logarithm:
t/τ = −ln(0.05) ≈ 3
Therefore:
t = 3 × 40 = 120 seconds
Option Analysis
Each option corresponds to a multiple of the time constant (τ = 40 s).
- (A) 60 s: Equals 1.5τ
Response ≈ 1 − e−1.5 ≈ 78% (below 95%) - (B) 80 s: Equals 2τ
Response ≈ 1 − e−2 ≈ 86% (still below 95%) - (C) 120 s: Equals 3τ
Response ≈ 1 − e−3 ≈ 95% ✅ - (D) 160 s: Equals 4τ
Response ≈ 1 − e−4 ≈ 98% (exceeds 95%, not minimal)
Summary Table
| Time (s) | Multiple of τ | % Steady-State | Correct? |
|---|---|---|---|
| 60 | 1.5 | 78% | No |
| 80 | 2 | 86% | No |
| 120 | 3 | 95% | Yes |
| 160 | 4 | 98% | No |
