help with math hw please?

Question

suppose hot chocoalte is poured at 70 degrees celcius and placed in a room at 25 degrees celcius. the hot chocolate cools by 10 degrees after one hour. how long after pouring the hot chocolate will it reach a temp of 40 degrees?

Answer 1

Let T(t) = temperature of the hot chocolate t hours after being poured

By Newton's Law of Cooling, T(t) = Ce^kt + 25

Since T(0) = 70, Ce⁰ + 25 = 70. So, C = 45

T(t) = 45e^kt + 25

Since T(1) = 60, 45e^k + 25 = 60

45e^k = 35

e^k = 7/9

k = ln(7/9) = -0.2513144

Therefore, T(t) = 45e^-0.2513144t + 25

So, if T(t) = 40, we have 45e^-0.2513144t + 25 = 40

45e^-0.2513144t = 15

e^-0.2513144t = 1/3

-0.2513144t = ln(1/3)

t = 4.37 hours (about 4 hours and 22 minutes)

Answer 2

(70-40)/10 = 30/10 = 3