Answer:
dp/dt = 10⁻⁵p(2225 − 4.2381p)
Step-by-step explanation:
Since the function is
0 = ln p + 45.8177 − ln (2225 − 4.2381p) − 0.02225t
Differentiating the function implicitly with respect to time, we have
d0/dt = d[ln p + 45.8177 − ln (2225 − 4.2381p) − 0.02225t]/dt
d0/dt = d[ln p]/dt + d[45.8177]/dt − d[ln (2225 − 4.2381p)]/dt − d[0.02225t]/dt
0 = (1/p)dp/dt + 0 − (1/(2225 − 4.2381p)) × -4.2381 dp/dt − 0.02225
0 = (1/p)dp/dt + 4.2381 dp/dt/(2225 − 4.2381p) − 0.02225
-(1/p)dp/dt - 4.2381 dp/dt/(2225 − 4.2381p) = − 0.02225
(1/p)dp/dt + 4.2381 dp/dt/(2225 − 4.2381p) = 0.02225
Factorizing out dp/dt from the left hand side, we have
[(1/p) + 4.2381/(2225 − 4.2381p)]dp/dt = 0.02225
taking L.C.M of the left-hand-side and simplifying, we have
[((2225 − 4.2381p + 4.2381p)/p(2225 − 4.2381p)]dp/dt = 0.02225
[2225/p(2225 − 4.2381p)]dp/dt = 0.02225
dividing both sides by2225, we have
[1/p(2225 − 4.2381p)]dp/dt = 0.02225/2225
[1/p(2225 − 4.2381p)]dp/dt = 0.00001
[1/p(2225 − 4.2381p)]dp/dt = 10⁻⁵
multiplying both sides by p(2225 − 4.2381p), we have
p(2225 − 4.2381p)[1/p(2225 − 4.2381p)]dp/dt = 10⁻⁵p(2225 − 4.2381p)
So, dp/dt = 10⁻⁵p(2225 − 4.2381p)
