The temperature T in #""^oC# of a particular city during a 24 hour period can be modelled by #T = 10 + 8 sin 12 pi t# where t is the time in hours, with t = 0 corresponding to midday. Find the rate at which the temperature is changing at 4pm.?

Question

Christopher Allers · Accepted Answer

#96^@C//hour# #T=10+8sin12pit# When it is #1200# time, #t=0#. When it is #1600# time, #t=4#. #(dT)/dt=96cos12pit# When #t=4#, #(dT)/dt=96cos48pi# #color(white)((dT)/dt)=96#

Scarlett Allison · Answer

undefined To find the rate at which the temperature is changing at 4 pm, we need to calculate the derivative of the temperature function T with respect to time t, and then evaluate it at t = 4.

Given T = 10 + 8 sin(12πt), we first find the derivative of T with respect to t:
dT/dt = 8(12π)cos(12πt)

Now, we evaluate this derivative at t = 4:
dT/dt = 8(12π)cos(12π*4)

dT/dt = 8(12π)cos(48π)

dT/dt ≈ 8(12π)(1) [since cos(48π) = 1]

dT/dt ≈ 96π

Therefore, the rate at which the temperature is changing at 4 pm is approximately 96π degrees Celsius per hour.