The temperature (in degrees Celsius) of two identical metal spheres is monitored over…

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

The temperature TT (in degrees Celsius) of two identical metal spheres is monitored over time tt (in hours). For each sphere, the rate of change of temperature satisfies dTdt=8−2t,0≤t≤6.\frac{dT}{dt} = 8 - 2t, \quad 0 \le t \le 6. Sphere A has T(0)=15T(0) = 15, and Sphere B has T(0)=22T(0) = 22. (1) Part 1: Find explicit formulas TA(t)T_A(t) and TB(t)T_B(t) describing the temperatures of Sphere A and Sphere B, respectively, for 0≤t≤60 \le t \le 6. (2) Part 2: Compute the difference TB(3)−TA(3)T_B(3) - T_A(3). (3) Part 3: Determine the time tt (if any, within the given interval) at which the temperature difference between the two spheres equals 5∘C5^\circ\text{C}. Show all reasoning. (4) Part 4: Express TB(t)T_B(t) algebraically in terms of TA(t)T_A(t) and a constant, and interpret this relationship in the context of the two physical experiments. (5) Part 5: Explain why the two spheres experience exactly the same instantaneous rate of temperature change at every moment even though their temperatures differ.

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