Please help me with this

Question

A population is growing according to P=P0e^0.09t. Find the doubling time. Round your answer to one decimal place.

years.

Answer 1

A population is growing according to P=P0e^0.09t. Find the doubling time. Round your answer to one decimal place.

The equation is exponential and increasing, because of the positive value for the rate, r.

The first thing you should do is make a list of what you know given the information in the problem.

P= The population after a given time, t, in years.

P₀ = The initial population

r = the rate of increase per year = 0.09, which is 9%

t = Time in years

e = a constant value ~2.78. It is not a variable. It doesn't change. It is just a number.

The key idea in this problem is that doubling time is the number of years it takes for the population to double. You don't need to know the initial population - you only need to know that P is going to be equal to 2*P₀. You substitute 2P₀ in for P and solve for t.

P = P₀•e^rt * Begin with the equation.

2P₀ = P₀•e^0.09t * Plug in known values

2P₀/P₀ = e^0.09t. * Isolate the exponential (get the e-term all by itself on one side of the equation) by dividing both sides by P_0. The P₀ terms cancel and only 2 remains on the left side of the equation.

2 = e^0.09t * The equation is in exponential form now, which is x = a^y, where 2 is x, e is a, and 0.09t is y. To solve for the t in the exponent, we switch the exponential form to logarithmic form. Logarithmic form is: y = log_a(x). Note that x is the argument of the logarithm, NOT multiplied to the logarithm. Now we literally switch forms, by placing x, a, and y in their spots in the logarithmic form equation.

x = a^y becomes y = log_a(x). *The rule for switching

2 = e^0.09t becomes 0.09t = log_e(2). * The switch

So now we have

0.09t = log_e(2). * We can simplify the right side by knowing that log_e(x) = ln(x) and we get

0.09t = ln(2). * Lastly, divide both sides by 0.09 to isolate t and get t's value.

t = ln(2)/0.09

t ~= 7.702 years.

So...the answer is:

At the exponentially increasing rate of 9% per year, every year, it will take approximately 7.7 years for the population to double.

So... Is this correct? Does this answer make sense? Let's check...

The population equation P =P₀ • e^(0.09t) is what we will use to see if we are correct, but this time...

P = The value we will solve for... But we know it will be twice the amount of P₀, so really, we are testing to see if we get 2000 for P.

P₀ = 1000. I just picked a number for our initial population. 1000 sounded good and so, it is!

r = 0.09 the given rate

t = 7.702 years (the double time)

We just plug all our numbers in and see if we get 2000 for P.

P =P₀ • e^(0.09t)

P = 1000 • e^{(0.09•7.702)} * Just input the entire right side of the equation into your calculator and get

P = 2000.066 *Close enough! So we conclude that our double time calculation is correct!

I hope it makes sense! Please ask if any part of my solution is unclear and I will answer as promptly as possible :)