Let p denote the population of the United States (in millions) in the year t, and assume that p is defined implicitly as a function of t by the equation
0 = ln p + 45.8177 − ln (2225 − 4.2381p) − 0.02225t.
Use implicit differentiation to show that the rate of change of p with respect
to t satisfies the equation:
dp
dt = 10−5p(2225 − 4.238p).

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)

One of the many reasons why Cortes and Pizarro were able to defeat the Aztecs and the Incas was because " d. The Europeans carried deadly diseases which decimated the populations of both the Aztecs and the Incas and thus undermined their political and military order," since disease actually killed more than weapons.

Explanation: