Solve the problem. The population of a particular city is...

Solve the problem. The population of a particular city is increasing at a rate proportional to its size. It follows the function $P(t) = 1 + ke^{0.12t}$ where $k$ is a constant and $t$ is the time in years. If the current population is 19,000, in how many years is the population expected to be 47,500? Round to the nearest year. 8 yr 3 yr 53 yr 5 yr

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00:01 Now in this problem we are given that the population of a particular city is increasing at a rate proportional to its size and we are also given with the function p of t.

00:13 This is equals to 1 plus k times e to the power 0 .04 t.

00:20 Now here k is a constant and t is the time in years.

00:27 Now then it says that if the current population is 21 ,000 that means if t is 0 the population is 21 ,000.

00:41 The current population.

00:42 Now our task is to tell that in how many years is the population expected to become 52 ,500.

00:51 This is what we have to find.

00:53 Now let's first find the value of this constant k by using the given data.

00:58 If you put t as 0 here, p of 0 is what? 21 ,000.

01:05 Right? plug it here.

01:08 So 21 ,000 is equals to 1 plus k is what we have to find.

01:14 E to the power 0 .04 multiplied with 0.

01:18 So this would be 1 plus k is equals to 21 ,000 and from here k would be equals to 20 ,999.

01:25 Now put the value of k in this function.

01:31 So p of t is equals to 1 plus k is 20 ,999 times e to the power 0 .04 t.

01:41 Now our task is to find t when the population becomes 52 ,500.

01:49 Right? so we can plug it here in this formula.

01:54 So p of t becomes 52 ,500 is equals to 1 plus 20 ,999 e to the power 0 .04 t...

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