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)

2 & 7

step-by-step explanation:

he climbs 300 ft/hr at a constant rate. the peak is at 1,500 ft.

when going up, he reaches 600 ft in 2 hours. remember, that he is going in a constant rate. divide 600 with 300:

600/300 = 2

assuming that he takes the same route back (meaning that it's a 1500 ft route back, you can solve.

note also that you are solving when he gets back to 600 ft after he reaches the peak.

first, find the amount of hours it takes to get up the peak. divide the total amount (1500), with the ft/her (300) to get the amount of hours (x):

1500/300 = x

x = 5

it takes jamie 5 hours to get up. now, he is returning down, and reaches 600 (his current destination). remember that he climbs at a constant 300 ft/hr. divide 600 with 300:

600/300 = x

x = 2

now add the two numbers together.

x = 5 + 2 = 7

jamie takes 7 hours to go up the mountain and reach his current position when descending.

(total amount/constant rate) = (time).

remember that it is a constant rate, so in total, when he gets back down the mountain, it will take the same amount of time as when he goes up. the total amount for the round trip is: 5 + 5 = 10.

~

where is the diagram? shown below where? ?

step-by-step explanation: