QUARTER FINAL STAGE 2026
Achimota School: 55 points
Adisadel College: 28 points
Boa Amponsem SHS: 23 points
QUESTION
An object at 47°C is in thermal contact with a heat bath at 27°C.
Assuming convection-dominated heat loss, find the rate of change of the object's temperature at 33°C if its initial rate of temperature change is $-15$ K/s.
ANSWER: $-4.5$ K/s
SOLUTION 1:
By Newton's law of cooling, the rate of temperature change is directly proportional to the temperature difference ($\dfrac{dT}{dt} \propto \Delta T$).
Initial difference: $\Delta T_1 = 47 - 27 = 20\text{ K}$
New difference: $\Delta T_2 = 33 - 27 = 6\text{ K}$
$\left(\dfrac{dT}{dt}\right)_2 = \left(\dfrac{dT}{dt}\right)_1 \times \dfrac{\Delta T_2}{\Delta T_1}$
$\left(\dfrac{dT}{dt}\right)_2 = -15 \times \dfrac{6}{20}$
$\left(\dfrac{dT}{dt}\right)_2 = -15 \times 0.3$
$\left(\dfrac{dT}{dt}\right)_2 = -4.5\text{ K/s}$
SOLUTION 2:
Using pure fractions to eliminate decimals from all steps:
$\Delta T_1 = 20\text{ K}$, $\Delta T_2 = 6\text{ K}$, $\left(\dfrac{dT}{dt}\right)_1 = -15\text{ K/s}$
$\left(\dfrac{dT}{dt}\right)_2 = -15 \times \dfrac{6}{20}$
Reduce the fraction by dividing the numerator and denominator by 2:
$\left(\dfrac{dT}{dt}\right)_2 = -15 \times \dfrac{3}{10}$
$\left(\dfrac{dT}{dt}\right)_2 = -\dfrac{45}{10}$
$\left(\dfrac{dT}{dt}\right)_2 = -4.5\text{ K/s}$
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PRACTICE QUESTIONS
1. QUESTION: An object at 49°C is in thermal contact with a heat bath at 25°C. Assuming convection-dominated heat loss, find the rate of change of the object's temperature at 31°C if its initial rate of temperature change is −16 K/s.
ANSWER: −4.0 K/s
SOLUTION:
$\dfrac{dT}{dt} \propto \Delta T$
$\Delta T_1 = 49 - 25$
$\Delta T_1 = 24\text{ K}$
$\Delta T_2 = 31 - 25$
$\Delta T_2 = 6\text{ K}$
$\left(\dfrac{dT}{dt}\right)_2 = \left(\dfrac{dT}{dt}\right)_1 \times \dfrac{\Delta T_2}{\Delta T_1}$
$\left(\dfrac{dT}{dt}\right)_2 = -16 \times \dfrac{6}{24}$
$\left(\dfrac{dT}{dt}\right)_2 = -\dfrac{6 \times 2}{3}$
$\left(\dfrac{dT}{dt}\right)_2 = -2 \times 2$
$\left(\dfrac{dT}{dt}\right)_2 = -4.0\text{ K/s}$
2. QUESTION: An object at 63°C is in thermal contact with a heat bath at 27°C. Assuming convection-dominated heat loss, find the rate of change of the object's temperature at 36°C if its initial rate of temperature change is −12 K/s.
ANSWER: −3.0 K/s
SOLUTION:
$\dfrac{dT}{dt} \propto \Delta T$
$\Delta T_1 = 63 - 27$
$\Delta T_1 = 36\text{ K}$
$\Delta T_2 = 36 - 27$
$\Delta T_2 = 9\text{ K}$
$\left(\dfrac{dT}{dt}\right)_2 = \left(\dfrac{dT}{dt}\right)_1 \times \dfrac{\Delta T_2}{\Delta T_1}$
$\left(\dfrac{dT}{dt}\right)_2 = -12 \times \dfrac{9}{36}$
$\left(\dfrac{dT}{dt}\right)_2 = -\dfrac{9}{3}$
$\left(\dfrac{dT}{dt}\right)_2 = -3.0\text{ K/s}$
3. QUESTION: An object at 55°C is in thermal contact with a heat bath at 23°C. Assuming convection-dominated heat loss, find the rate of change of the object's temperature at 31°C if its initial rate of temperature change is −18 K/s.
ANSWER: −4.5 K/s
SOLUTION:
$\dfrac{dT}{dt} \propto \Delta T$
$\Delta T_1 = 55 - 23$
$\Delta T_1 = 32\text{ K}$
$\Delta T_2 = 31 - 23$
$\Delta T_2 = 8\text{ K}$
$\left(\dfrac{dT}{dt}\right)_2 = \left(\dfrac{dT}{dt}\right)_1 \times \dfrac{\Delta T_2}{\Delta T_1}$
$\left(\dfrac{dT}{dt}\right)_2 = -18 \times \dfrac{8}{32}$
$\left(\dfrac{dT}{dt}\right)_2 = -\dfrac{18}{4}$
$\left(\dfrac{dT}{dt}\right)_2 = -\dfrac{9}{2}$
$\left(\dfrac{dT}{dt}\right)_2 = -4.5\text{ K/s}$
4. QUESTION: An object at 69°C is in thermal contact with a heat bath at 21°C. Assuming convection-dominated heat loss, find the rate of change of the object's temperature at 33°C if its initial rate of temperature change is −24 K/s.
ANSWER: −6.0 K/s
SOLUTION:
$\dfrac{dT}{dt} \propto \Delta T$
$\Delta T_1 = 69 - 21$
$\Delta T_1 = 48\text{ K}$
$\Delta T_2 = 33 - 21$
$\Delta T_2 = 12\text{ K}$
$\left(\dfrac{dT}{dt}\right)_2 = \left(\dfrac{dT}{dt}\right)_1 \times \dfrac{\Delta T_2}{\Delta T_1}$
$\left(\dfrac{dT}{dt}\right)_2 = -24 \times \dfrac{12}{48}$
$\left(\dfrac{dT}{dt}\right)_2 = -\dfrac{12}{2}$
$\left(\dfrac{dT}{dt}\right)_2 = -6.0\text{ K/s}$
5. QUESTION: An object at 43°C is in thermal contact with a heat bath at 27°C. Assuming convection-dominated heat loss, find the rate of change of the object's temperature at 39°C if its initial rate of temperature change is −8.0 K/s.
ANSWER: −6.0 K/s
SOLUTION:
$\dfrac{dT}{dt} \propto \Delta T$
$\Delta T_1 = 43 - 27$
$\Delta T_1 = 16\text{ K}$
$\Delta T_2 = 39 - 27$
$\Delta T_2 = 12\text{ K}$
$\left(\dfrac{dT}{dt}\right)_2 = \left(\dfrac{dT}{dt}\right)_1 \times \dfrac{\Delta T_2}{\Delta T_1}$
$\left(\dfrac{dT}{dt}\right)_2 = -8.0 \times \dfrac{12}{16}$
$\left(\dfrac{dT}{dt}\right)_2 = -\dfrac{12}{2}$
$\left(\dfrac{dT}{dt}\right)_2 = -6.0\text{ K/s}$
6. QUESTION: An object at 47°C is in thermal contact with a heat bath at 22°C. Assuming convection-dominated heat loss, find the rate of change of the object's temperature at 37°C if its initial rate of temperature change is −15 K/s.
ANSWER: −9.0 K/s
SOLUTION:
$\dfrac{dT}{dt} \propto \Delta T$
$\Delta T_1 = 47 - 22$
$\Delta T_1 = 25\text{ K}$
$\Delta T_2 = 37 - 22$
$\Delta T_2 = 15\text{ K}$
$\left(\dfrac{dT}{dt}\right)_2 = \left(\dfrac{dT}{dt}\right)_1 \times \dfrac{\Delta T_2}{\Delta T_1}$
$\left(\dfrac{dT}{dt}\right)_2 = -15 \times \dfrac{15}{25}$
$\left(\dfrac{dT}{dt}\right)_2 = -\dfrac{3 \times 15}{5}$
$\left(\dfrac{dT}{dt}\right)_2 = -3 \times 3$
$\left(\dfrac{dT}{dt}\right)_2 = -9.0\text{ K/s}$
7. QUESTION: An object at 62°C is in thermal contact with a heat bath at 26°C. Assuming convection-dominated heat loss, find the rate of change of the object's temperature at 53°C if its initial rate of temperature change is −12 K/s.
ANSWER: −9.0 K/s
SOLUTION:
$\dfrac{dT}{dt} \propto \Delta T$
$\Delta T_1 = 62 - 26$
$\Delta T_1 = 36\text{ K}$
$\Delta T_2 = 53 - 26$
$\Delta T_2 = 27\text{ K}$
$\left(\dfrac{dT}{dt}\right)_2 = \left(\dfrac{dT}{dt}\right)_1 \times \dfrac{\Delta T_2}{\Delta T_1}$
$\left(\dfrac{dT}{dt}\right)_2 = -12 \times \dfrac{27}{36}$
$\left(\dfrac{dT}{dt}\right)_2 = -\dfrac{27}{3}$
$\left(\dfrac{dT}{dt}\right)_2 = -9.0\text{ K/s}$
8. QUESTION: An object at 42°C is in thermal contact with a heat bath at 24°C. Assuming convection-dominated heat loss, find the rate of change of the object's temperature at 36°C if its initial rate of temperature change is −9.0 K/s.
ANSWER: −6.0 K/s
SOLUTION:
$\dfrac{dT}{dt} \propto \Delta T$
$\Delta T_1 = 42 - 24$
$\Delta T_1 = 18\text{ K}$
$\Delta T_2 = 36 - 24$
$\Delta T_2 = 12\text{ K}$
$\left(\dfrac{dT}{dt}\right)_2 = \left(\dfrac{dT}{dt}\right)_1 \times \dfrac{\Delta T_2}{\Delta T_1}$
$\left(\dfrac{dT}{dt}\right)_2 = -9.0 \times \dfrac{12}{18}$
$\left(\dfrac{dT}{dt}\right)_2 = -\dfrac{12}{2}$
$\left(\dfrac{dT}{dt}\right)_2 = -6.0\text{ K/s}$
9. QUESTION: An object at 92°C is in thermal contact with a heat bath at 28°C. Assuming convection-dominated heat loss, find the rate of change of the object's temperature at 44°C if its initial rate of temperature change is −24 K/s.
ANSWER: −6.0 K/s
SOLUTION:
$\dfrac{dT}{dt} \propto \Delta T$
$\Delta T_1 = 92 - 28$
$\Delta T_1 = 64\text{ K}$
$\Delta T_2 = 44 - 28$
$\Delta T_2 = 16\text{ K}$
$\left(\dfrac{dT}{dt}\right)_2 = \left(\dfrac{dT}{dt}\right)_1 \times \dfrac{\Delta T_2}{\Delta T_1}$
$\left(\dfrac{dT}{dt}\right)_2 = -24 \times \dfrac{16}{64}$
$\left(\dfrac{dT}{dt}\right)_2 = -\dfrac{24}{4}$
$\left(\dfrac{dT}{dt}\right)_2 = -6.0\text{ K/s}$