#(dL)/dt = KL^2 ln t, L(1)=-5#
#(dL)/dt = KL^2 ln t #
#(dL)/L^2 = K ln t dt#
#int(dL)/L^2 = K ln t dt#
#-1/L= K[tln(t)-t]+C#
#-1/-5= K[tln(1)-1]+C#
#1/5= -K+C#
#C= 1/5 + K#
#-1/L= K[tln(t)-t]+1/5 + K#
#color(orange)(L= -1/(K[tln(t)-t]+1/5 + K))#
#color(red)(a))#
Due to Newton's law of cooling: #(dT)/dt = k(T-75)#
#(dT)/dt = k(T-75)#
#y(t) = T(t) - 75#
#y(0) = T(0) - 75#
#y(0) = 185 - 75#
#y(0) = 110#
#(dy)/dt = ky#
#y(t) = 110e^(kt)#
#y(30) = 110e^(30t)#
#y(30)# because it is going in the oven for half an hour
#150-75 = 110e^(30t)#
#75 = 110e^(30t)#
#75/100 = e^(30t)#
#ln(75/110) = 30t#
#t= (ln(75/110)/30) #
#y(t) = 110e^(tln(75/110)/30)#
#y(45) = 110e^(45(1/30)ln(75/110)#
#= 62#
#y(45) = 62+ 75#
#color(orange)(y(45) = 137^@ F)#
#color(red)(b))#
#y(t) = T(t) - 75#
#y(110) = 110 - 75=35#
#35 = 110e^(tln(75/110)/30)#
#35/110 = e^(tln(75/110)/30)#
#ln(35/110) = tln(75/110)/30#
#30ln(35/110) = tln(75/110)#
#(30ln(35/110))/ln(75/110) = t#
#color(orange)(t=90 min#
#color(red)(a))#
#p(t)=100e^(kt)#
#290=100e^(k(1))#
#2.9=e^(k)#
#ln2.9=k#
#p(t)=100e^(ln(2.9)t)#
#color(orange)(p(t)=100(2.9)^t)#
#color(red)(b))#
#p(4)=100(2.9)^4#
#color(orange)(=7072.81)# bacteria
#color(red)(c))#
#p(t)=100e^(kt)# take derivative of this
#(d(p(t)))/dt=100((d)/dt)e^(kt)#
#p'(t)=100ke^(kt)#
#ln2.9=k#
#p'(t)=100(ln2.9)e^(ln2.9t)#
#p'(t)=100(ln2.9)(2.9^t)#
#p'(4)=100(ln2.9)(2.9^4)#
#color(orange)(p'(4)= 7530.50)#
#color(red)(d))#
#10000=100(2.9)^t#
#100=(2.9)^t#
#ln100=tln(2.9)#
#t= ln100/ln(2.9)#
#color(orange)(t= 4.3 hrs)#
