An experiment is conducted in which the temperature (T) in degrees Celsius (°C) varies with time (t) in seconds according to the formula: $$T(t) = 37.5 + 7t - 0.5t^2$$ where $0 \leq t \leq 10$ 8.1 Write down the initial temperature - NSC Technical Mathematics - Question 8 - 2022 - Paper 1

To determine the rate of change of temperature, we find the derivative of the temperature function:

$T'(t) = 7 - t$

Substituting $t = 4$:

$T'(4) = 7 - 4 = 3$

Hence, the rate of change of temperature at $t = 4$ seconds is 3 °C/s.

To find the maximum temperature, we find the time when $T'(t) = 0$:

$7 - t = 0 \Rightarrow t = 7$

Now, we evaluate $T(t)$ at $t = 7$:

$T(7) = 37.5 + 7(7) - 0.5(7)^2 = 62$

Thus, the maximum temperature reached during the experiment is 62 °C.

The temperature is decreasing when $T'(t) < 0$:

$T'(t) = 7 - t < 0 \Rightarrow t > 7$

Therefore, the temperature is decreasing during the interval $(7, 10]$.