Hey, I on log and continuous growth.

[firstwave]

Hey, here is a question I couldn't figure out. I think it's actually really easy, but the wording confuses me.

A recent survey showed that the population, P, of Victoria is growing at an annual rate of 1.1%. Let P0 represent the population on January 1, 2001 and let t represent the time, in years, since this date.

Question:
Express P as a function of t in the form P = P0 a^t

Thanks.

 

[HallsofIvy]

You want P= P₀a^t. Since P and t are variables, the only problem is finding the constants P₀ and a. You are told to "Let P0 represent the population on January 1, 2001" and aren't told what that is. If you are not given that as a specific number, and can't look it up the only thing you can do is leave like that: P₀.
You are told that "the population, P, of Victoria is growing at an annual rate of 1.1%". Okay, if the the population in 2001 was P₀ then the population in 2002 (one year later so t=1) had increased by 1.1%:P(1) was P₀+ 1.1% of P₀= P₀+ 0.011P₀= 1.011P₀.
Now put that into your equation, P= P₀a^t:
1.011P₀= P₀a¹ and solve for a.
(It's not very hard!)

 

[quicknote]

Hi there,

Thank you for reaching out and asking about this question. I am happy to help clarify the concept of log and continuous growth.

Log growth refers to a type of growth that follows a logarithmic function, where the growth rate decreases over time. On the other hand, continuous growth refers to a type of growth that follows an exponential function, where the growth rate remains constant over time.

Regarding the question you have mentioned, the wording may seem confusing at first, but it is actually a simple concept. The key is to understand the variables and their meanings.

In this case, P represents the population, P0 represents the initial population on January 1, 2001, and t represents the time since this date in years. The annual growth rate of 1.1% means that the population is increasing by 1.1% every year.

To express P as a function of t in the form P = P0 a^t, we can use the formula for continuous growth, which is P = P0e^(rt), where e is the mathematical constant approximately equal to 2.71828, and r is the growth rate in decimal form.

In this case, r = 0.011 (1.1% expressed in decimal form), and we can plug it into the formula to get P = P0e^(0.011t).

I hope this helps clarify the concept of log and continuous growth and how to express P as a function of t in the given form. If you have any further questions, please feel free to ask. Good luck with your studies!