To determine the gain of the transistor amplifier at 28.0°C, given that the gain changes linearly with temperature, we can use the information provided about the gain at two specific temperatures: 20.0°C and 55.0°C. We know that at 20.0°C, the gain is 30.0, and at 55.0°C, the gain is 35.2. Let's break this down step by step.
Understanding Linear Relationships
When we say that the gain depends linearly on temperature, it means that we can express the gain as a linear function of temperature. This can be represented by the equation:
Gain(T) = m * T + b
Where:
- Gain(T) is the gain at temperature T.
- m is the slope of the line (change in gain per degree change in temperature).
- b is the y-intercept (gain when temperature is zero, though we won't need this value directly).
Calculating the Slope (m)
To find the slope (m), we can use the two points we have:
- Point 1: (20.0°C, 30.0)
- Point 2: (55.0°C, 35.2)
The formula for the slope between two points is:
m = (Gain2 - Gain1) / (T2 - T1)
Substituting the values:
m = (35.2 - 30.0) / (55.0 - 20.0) = 5.2 / 35.0 ≈ 0.1486
Finding the Gain at 28.0°C
Now that we have the slope, we can find the gain at 28.0°C. We can use the point-slope form of the linear equation. We can choose either of the two points to find the y-intercept (b), but let's use the first point (20.0°C, 30.0):
30.0 = 0.1486 * 20.0 + b
Calculating b:
b = 30.0 - (0.1486 * 20.0) = 30.0 - 2.972 ≈ 27.028
Now we have the complete linear equation:
Gain(T) = 0.1486 * T + 27.028
Next, we substitute T = 28.0°C into the equation:
Gain(28.0) = 0.1486 * 28.0 + 27.028
Calculating this gives:
Gain(28.0) ≈ 4.1648 + 27.028 ≈ 31.193
Final Result
Therefore, the gain of the transistor amplifier at 28.0°C is approximately 31.2. This demonstrates how temperature can affect the performance of electronic components, and understanding these relationships is crucial in designing reliable circuits.
