pre- calculus exponents and logarithms

Question

Robert takes a roast out of the oven when the internal temperature of the roast is 165°F. After 15 minutes, the temperature of the roast drops to 135°F.

The temperature of the room is 70°F.

How long does it take for the temperature of the roast to drop to 110°F?

Use the Newton's Law of Cooling equation 

Round your answer to the nearest minute.

Answer 1

Newton's Law of Cooling: T = (T₀ - T_A)e^-kt + T_A or T₀ + T_A(1 - e^-kt)

The 1st part allows you to solve for k by pugging in T₀=165 and T_A= 70 T(t = 15) = 135

Rearranging the equation, you get (T-T_A)/(T₀-T_A) = e^-kt

Take the ln of both sides and rearranging, you get -1/t ln[(T-T_A)/(T₀-T_A)] = k

How long to drop to 110 under the same conditions?

That's finding t knowing T's and k from previously. From the last equation, you can just switch k and t and plug in the new T to solve.