Complete question :
A welder drops a piece of red-hot steel on the floor. The initial temperature of the steel is 2,500 degrees Fahrenheit. The
ambient temperature is 80 degrees Fahrenheit. After 2 minutes the temperature of the steel is 1,500 degrees. The function
f(t) = Ce^-kt + 80 represents the situation, where t is time in minutes, C is a constant, and k is a constant.
After 2 minutes the temperature of the steel is 1,500 degrees. After how many minutes will the temperature of the steel be
100 degrees and therefore safe to pick up with bare hands? Round your answer to the nearest whole number, and do not.
include units.
Answer:
Kindly check explanation
Step-by-step explanation:
Given the exponential function :
f(t) = Ce^(-kt) + 80
t = time in minutes ; k = constant ; C = constant; Initial temperature = 2500 = temperature at t = 0
Hence,
2500 = Ce^k*0 + 80
2500 = C + 80
C = 2500 - 80
C = 2420
Temperature after 2 minutes ; t = 2 minutes
1500 = 2420e^-k*2 + 80
1500 = 2420e^-2k + 80
1500 - 80 = 2420e^-2k
1420 = 2420e^2k
1420 / 2420 = e^-2k
0.58677 = e^-2k
Take the In of both sides
In(0.58677) = - 2k
−0.533122 = - 2k
k = 0.533122 / 2
k = 0.2665
k = 0.2667
Time at which temperature falls to 100
f(t) = Ce^(-kt) + 80
100 = 2420 * e^(-0.267t) + 80
100 - 80 = 2420 * e^(-0.267t)
20 = 2420 * e^(-0.267t)
Divide both sides by 2420
0.0082644 = e^(-0.267t)
Take In of both sides
−4.795790 = - 0.267t
Divide through by - 0.267
t = 17.96176
t = 17.96 minutes
