I have a maths assignment and i need help with this question my teacher is overseas at the moment and it is due when he gets back

The Population P, of penguins in a certain region of Antarctica can be modeled by the relation P=Po (2) t/90, where t is time, measured in months and Po is the initial number of penguins.

a) if there are 800 penguins in the region today, how many will there be in 2 years?

b) what does the value of 90 represent in this formula?

c) How long will it take before there are 12 800 penguins?

this is only the first question on the sheet so i need the working out because then i will be able to do the rest myself

thank you if you can help

Exponential functions are indicated on typographical material as a small number written on the top right of the base, like:

2t/90.
While this is relatively difficult to do in a post, we usually represent the same expression using straight typing as:
2^(t/90)
Do not forget the parentheses, because exponentiation takes precedence over multiplication/division.

So please check that the given formula is:
P=Po (2)^(t/90)

a.
Po=population today, so Po=800.
t=number of months, so t=24 for two years.
There leaves only one unknown,
P=800*(2)^(24/90)=962.4, say 963

b. 90 represents a factor for the rate of exponential growth. The bigger the factor, the slower the population grows.

c. "How long" means we need to find "t".
So
12800=800*(2)^(t/90)
12800/800=(2)^(t/90)
16=2^(t/90)
take log to the base 2:
log216=(t/90)
4=t/90
so t=4*90=360 months = 20 years, assuming the population grows unchecked.

Sure, I'll be happy to help you solve these math problems regarding the penguin population in Antarctica.

a) To find the number of penguins in 2 years, we need to substitute the values into the formula. In the given formula, P represents the population, Po represents the initial number of penguins, t represents time in months, and 90 is a constant.

Given:
P = 800 (current population)
t = 2 (time in years)

To convert years to months, we need to multiply by 12 since there are 12 months in a year. Therefore, the time in months is t = 2 x 12 = 24 months.

Substituting the values into the formula: P = Po(2)^(t/90)
800 = Po(2)^(24/90)

Now, we need to solve for Po (the initial number of penguins).
To isolate Po, we divide both sides of the equation by (2)^(24/90):

800 / (2)^(24/90) = Po

Calculating this expression will give you the initial number of penguins in the region 2 years ago.

b) The value of 90 represents the time, in months, at which the population will double. In other words, after every 90 months, the penguin population will double because the base of the exponent (2) is raised to the power of t/90.

c) To determine how long it will take before there are 12,800 penguins, we need to substitute the values into the formula.

Given:
P = 12,800
Po = unknown (initial number of penguins)
t = unknown (time in months)

Substituting these values into the formula: 12,800 = Po(2)^(t/90)

We want to solve for t, so we need to isolate t.
To do this, we can take the logarithm of both sides of the equation.

log(12,800) = log(Po(2)^(t/90))

Since log(Po) is a constant, we can simplify the equation as follows:

log(12,800) = log(Po) + log((2)^(t/90))

Using logarithmic properties, we know that log(a^b) = b * log(a). Hence:

log(12,800) = log(Po) + (t/90) * log(2)

Let's assign a variable to the term log(Po), for example, let log(Po) = C. Now we have:

log(12,800) = C + (t/90) * log(2)

By rearranging the equation, we can solve for t:

t = (90 / log(2)) * (log(12,800) - C)

Substituting this equation into an expression to calculate the value of t will determine the amount of time it will take before there are 12,800 penguins in the region.

Remember to calculate log(12,800) and log(2) using a calculator based on the logarithm base (common logarithm, natural logarithm, etc.) your math assignment requires.

I hope this helps you solve your math assignment! If you have any further questions, please feel free to ask.