Solve.

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 15,000, in how many years is the population expected to be 37,500? (Round to the nearest year.)

solve

37500 = 1 + 15000(e)^.12t

(the 1 is rather ineffectual in this equation. Are you sure you typed it right way? )

Anyway, assuming it is correct

= 36499 = 15000(e)^.12t
2.4332667 = e^.12t
.12t = ln 2.4332667
.12t = .889235
t = 7.41 yrs or
in appr 7 years

To find the number of years it takes for the population to reach 37,500, we can set up an equation using the given function:

P(t) = 1 + ke^(0.12t)

We know that the current population is 15,000, so we can substitute P(t) = 15,000 and solve for t:

15,000 = 1 + ke^(0.12t)

Next, we can find the value of k by using the additional information that the population is increasing at a rate proportional to its size. This means that the derivative of P(t) should be proportional to P(t). Taking the derivative of P(t) with respect to t, we get:

P'(t) = 0.12ke^(0.12t)

We can see that P'(t) is proportional to ke^(0.12t) since the constant of proportionality is 0.12k. Now we can substitute P(t) = 15,000 and P'(t) = 0.12ke^(0.12t) into the equation:

0.12ke^(0.12t) = 15,000

Simplifying this equation, we get:

ke^(0.12t) = 15,000 / 0.12

ke^(0.12t) = 125,000

Now, we can substitute this value of k into the original equation:

15,000 = 1 + (125,000)e^(0.12t)

Simplifying further:

(125,000)e^(0.12t) = 15,000 - 1
(125,000)e^(0.12t) = 14,999

Now, divide both sides of the equation by 125,000:

e^(0.12t) = 14,999 / 125,000

Using natural logarithm, we can take the ln of both sides:

ln(e^(0.12t)) = ln(14,999 / 125,000)

Simplifying:

0.12t = ln(14,999 / 125,000)

Dividing both sides by 0.12:

t = ln(14,999 / 125,000) / 0.12

Using a calculator to approximate the value, we find:

t ≈ 22.073

Therefore, the population is expected to be 37,500 in approximately 22 years.

To find the number of years in which the population is expected to be 37,500, we need to solve the equation P(t) = 37,500 for t.

The given population function is P(t) = 1 + ke^(0.12t), where P(t) represents the population at time t and k is a constant.

We are given that the current population is 15,000, so we can substitute P(0) = 15,000 into the equation to find the value of k.

P(0) = 1 + ke^0
15,000 = 1 + k

Simplifying the equation, we have:
k = 15,000 - 1
k = 14,999

Now we substitute the value of k into the population function to solve for t:

37,500 = 1 + 14,999e^(0.12t)

Subtracting 1 from both sides, we get:
37,499 = 14,999e^(0.12t)

Dividing both sides by 14,999, we have:
2.50 = e^(0.12t)

To solve for t, we take the natural logarithm of both sides:
ln(2.50) = ln(e^(0.12t))

Using the property ln(e^x) = x, we simplify to:
ln(2.50) = 0.12t

Now we can solve for t by dividing both sides by 0.12:
t = ln(2.50) / 0.12

Using a calculator, we find:
t ≈ 7.414

To round to the nearest year, we can round up or down based on the decimal value. Since 0.414 is closer to 0, we round down.

Therefore, the population is expected to be 37,500 in approximately 7 years.