A weekly census of the tree frog in a park is given

week population
1 18
2 54
3 162
4 486
5 1458
6 4374

a. find the function of the form f(x)=Pa^x that describes the frog population at time x weeks.

b. what's the growth factor in this situation? (that is, by waht number must this week's population be multiplied by to obtain next weeks population?)

c. each tree frog requires 10 squre feet of space and the park has an area of 6.2 square miles will the space required by the frog population exceed the size of the park in 12 weeks? in 14?

Andy, I have done quite a few for you now.

Give this one a try, let me know what you get.

a. f(x)=6(3)^x

b.3
c. no, yes

a. To find the function that describes the frog population at time x weeks, we can use the given information and the form f(x) = Pa^x.

Let's take week 1 as our initial time, so x = 1 and the population is 18.

We can substitute this information into the form and solve for P and a:

18 = Pa^1
18 = P * a

Now let's take another week, for example, week 2:

54 = Pa^2

We can divide the second equation by the first equation to eliminate P:

54 / 18 = (Pa^2) / (Pa^1)

3 = a

Knowing that "a" is 3, we can substitute it back into the original equation to find P:

18 = P * 3^1
18 = P * 3
P = 6

Therefore, the function that describes the frog population at time x weeks is f(x) = 6 * 3^x.

b. The growth factor is the number by which this week's population must be multiplied to obtain next week's population. In this case, it is the constant "a" in the equation f(x) = Pa^x.

From part a, we found that a = 3. Therefore, each week's population is multiplied by 3 to obtain the next week's population.

c. To determine if the space required by the frog population will exceed the size of the park, we need to calculate the total space required by the frogs.

We know that each tree frog requires 10 square feet of space.

Let's calculate the space required by the frog population after 12 weeks and check if it exceeds the size of the park:

Population after 12 weeks = f(12) = 6 * 3^12

Total space required after 12 weeks = Population after 12 weeks * 10 square feet

Let's substitute the values and calculate:

Total space required after 12 weeks = 6 * 3^12 * 10 square feet

Now, the park has an area of 6.2 square miles. We need to convert this area into square feet to compare it with the total space required:

1 square mile = 27,878,400 square feet

Park size = 6.2 square miles * 27,878,400 square feet

Now we can compare the total space required after 12 weeks with the park size:

Is (Total space required after 12 weeks) > (Park size)?

If the total space required after 12 weeks exceeds the park size, then the space required by the frog population will exceed the size of the park. Repeat the same process for the 14th week to find if the space required by the frog population will exceed the park size in that time frame as well.