Question

7. The temperature of a Starbuck's coffee after n minutes is given by T = 72(0.85)" + 20. How long does it take for the coffee to cool to a temperature of 70°?

Answer 1

given, temperature after n minutes to be

so, at time n=0, that is when coffe was taken it's temeperature was,

Let's assume after 'x' min, temp comes down to 70. replacing in equation of temp gives,

$\Rightarrow 70 -20 = 72(0.85)^x$

$\Rightarrow 50 = 72(0.85)^x$

$\Rightarrow (0.85)^x=50/72$

$\Rightarrow (0.85)^x=0.6944$

to find x, take logarithm on both sides

$\Rightarrow xlog(0.85)=log(0.6944)$$\Rightarrow x=log(0.6944)/log(0.85)$ here,

$log(0.85)=-0.07058$ and

$log(0.6944)=-0.15836$

$\Rightarrow x=-0.15836/-0.07058=2.24=2+0.24$

it'll take 2.24 min for coffe to cool down to 70. but

and

so it'll take 2 min and 14.4 sec to cool coffe to 70deggre