Q3 3.1

P = 500 (E^( k t))
P[10] ==500 (E^( k 10))== 1.15*500

Solve[500(E^( k 10))==1.15*500]

P[t_]:= 500(E^( 0.013976194237515869` t))
{P[30] , P'[30]}
Plot[{650,P[t]}, {t, 0, 30}]
FindRoot[650==P[t], {t, 17}]

Q16 3.1

T’ = k (T-(Subscript[T, m])) 
Note (Subscript[T, m]) changes from 0 to 100 in this problem at t=2 
T(0) = 100 
T(1) = 90
T = 100 (E^( k t))  for 0<t<2

Solve[ 100 (E^(k 1  ))==90.,k]

P[2] = 100 (E^(--0.1053605156578263` *  2))= 81
P[t]= 100-19 (e^(k2 (-2+t)))  t > 2

DSolve[ {
T'[t]==k2 (T[t]-100), 
T[2]==81
},
T[t], t]//Simplify

Solve[ 
100-19 (E^(k2 (-2+3)))== 81. + 10, k2]

-0.1053605156578263`

Show[
Plot[{100-19 (E^(-0.747214401830221` (-2+t))),99.9},{t, 2, 5}],
Plot[ {100 (E^(-0.1053605156578263`t)),99.9},{t, 0, 2},
PlotRange->All],
PlotRange->{All,All}
]

Solve[ {100-19 (E^(-0.747214401830221` (-2+t))),99.9}==99.9, t]

DSolve[{
T'[t]== k (T[t] - 100),
T[2]== 81
},
T[t],
t]//Simplify

DSolve[ T'[t]==k (T[t]-0), T[t], t]