The temperature T(t) of a body at time t=0 is 160° F and it decreases continuously as per the differential equation (d T)/(d t)=-K(T-80), where K is a positive constant. If T(15)=120° F, then T(45) is equal to

Solution

Maths sir · 5 steps

Watch the gap above 80°F, not the temperature itself: it halves every 15 minutes.

1. Separate variables. Put every T on one side: dT/(T-80) = -K dt. Integrating gives ln(T-80) = -Kt + C.
2. Use the start. At t = 0 the gap is T - 80 = 160 - 80 = 80, so the solution is

   T - 80 = 80 e^(-Kt)

3. Use T(15). At t = 15: 120 - 80 = 40 = 80 e^(-15K), so e^(-15K) = 1/2. The gap halves every 15 minutes.
4. Find T(45). e^(-45K) = (e^(-15K))³ = 1/8, so the gap is 80 × 1/8 = 10. Then T(45) = 80 + 10 = 90.
5. So. T(45) = 90°F, which is option (a).

The idea to keep. For dT/dt = -K(T - T₀), the gap T - T₀ decays exponentially. Find the factor for one time interval, then raise it to a power. You do not need K itself.

Answer from the NTA final answer key. Solution by PrepWiser.