400et/2
600et/2
400e-t/2
300e-t/2
dp/dt = p/2
dp/p = (1/2)dt
ln|p| = t/2 + c
At t=0, p=100
ln|100| = c
ln|p| = t/2 + ln|100|
ln|p| = t/2 + ln(100)
ln|p| - ln(100) = t/2
ln|p/100| = t/2
|p/100| = et/2
p/100 = ±et/2
p = ±100et/2
Since p must be positive, p = 100et/2
If p(0) = 100, then p(t) = 100et/2. However, this is not one of the options.
Let's solve it another way:
dp/dt = p/2
dp/p = dt/2
Integrating both sides:
∫dp/p = ∫dt/2
ln|p| = t/2 + C
p = et/2 + C = Aet/2 (where A = eC)
Given p(0) = 100:
100 = Ae0 = A
Therefore, p(t) = 100et/2
This is not in the options. Let's re-examine the options.
Let's try separating the variables and integrating again:
∫dp/p = ∫(1/2)dt
ln|p| = t/2 + C
|p| = e(t/2 + C) = eCet/2 = Aet/2
If p(0) = 100, then 100 = Ae0 = A, so p(t) = 100et/2
Let's assume there's a mistake in the options. The correct solution is 100et/2