Question

After heating up in a teapot, a cup of hot water is poured at a temperature of 204°F.

Answer 1

Answer:

• Question 1: 0.081 min⁻¹

• Question 2: 168ºF

Explanation:

The missing equation is:

             $T=T_a+(T_0-T_a)e^{-kt}$

You are given:

• Ta = the temperature surrounding the object = 73ºF
• T₀ = the initial temperature of the object = 204ºF
• t = 2.5 min
• T after 2.5 min = 180ºF

Question 1: Find the the value of k.

Substitute the data into the equation and solve for k:

          $180\º=73\º+(204\º-73\º)e^{-k\cdot 2.5min}\\ \\ 107\º=131\ºe^{-k\cdot 2.5min}\\\\-k\cdot 2.5min=\ln (107/131)\\\\k=0.081min^{-1}$

Question 2: Find the Fahrenheit temperature of the cup of water, to the nearest degree, after 4 minutes.

Subsititute into the formula and compute:

            $T=73\ºF+(204\ºF-73\ºF)e^{-0.081min^{-1}\times 4min}\\ \\ T=73\ºF+(131\ºF)(0.72325024)\\\\T=168\ºF$