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).

Question

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Respuesta :

oladayoademosu oladayoademosu · Answer 1 · 2021-07-24T08:34:36+02:00

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)