Question

Please let me know if I have done everything correctly, especially 2nd paragraph and then I need help and explanation on the last paragraph. TIA

Answer 1

Data
Months (x) Population P(x)
0 5
1 10
2 20
3 40

This exponential function measures growth monthly. Using complete sentences, how would I explain how to find the rate of growth every week.

Originally, I thought I would calculate the Months (x) out to 12, which would be 20,480 and then divide that by 52, but the more I think about it, the more I am unconvinced that I am right.

Next, I have to calculate when the population will reach 500, which would be

[\frac{ 500 }{ 5 }=\frac{ 5(2^{x}) }{ 5 }\]\[100=2^{x}\]log(100) = (log)2^2\
log(100) = x (log)2
log(100)/log(2) = x
x = 6.64385619

Last, the system I have to put the function into only works in logarithms of base 10. Using complete sentences, how would I explain converting my exponential function P(x) in a logarithmic one and then into a base 10 logarithm.

This one I need help on.

Answer 2

\[\frac{ 500 }{ 5 }=\frac{ 5(2^{x}) }{ 5 }\]

Answer 3

Sorry... that what the equation should have been

Answer 4

Looking at the data we notice the population doubles every month.
So the exponential function is 2^x.
P(x) = A * 2^x
when x = 0, P = 5
5 = A * 2^0 = A * 1. So A = 5
P(x) = 5 * 2^x where P is the population in month x.

Answer 5

If we want the Population as a function of w (where w represents week) then:
P(w) = 5 * 2^(w/4) assuming 4 weeks in a month.

Answer 6

P(x) = 5 * 2^x
When will the population reach 500?
500 = 5 * 2^x
2^x = 100
Take logarithm to the base 10 on both sides:
x * log(2) = log(100) = 2
x = 2 / log(2) = 6.64 months

Answer 7

I started there, but because there really isn't 4 weeks in a month, I went to the 52 weeks

Answer 8

So I was correct on that part

Answer 9

The 6.64 months

Answer 10

Yes. The population will reach 500 in 6.64 months.

Answer 11

Okay, but what about the rate of growth every week. Wouldn't you have to calculate the growth at month 12 and then divide by 52 weeks?

Answer 12

This month to week conversion is an approximate one because months have: 28-31 days and the growth in month February will not be the same as the growth in March.

But going the year route may be better than the 4 weeks to a month route.

Answer 13

So would I be correct in my answer?

Answer 14

let me see.
52 weeks = 12 months
1 week = 12 / 52 months = 3/13 month

P(x) = 5 * 2^x (where x is in months) OR
P9w) = 5 * 2^(3w/13) (where w is in weeks)

Answer 15

P(w) = 5 * 2^(3w/13) (where w is in weeks)

Answer 16

Instead of dividing w by 4 to convert it to a month we are dividing w by 4.33333 to convert it to an average month.

Answer 17

1 week is 3/13th of a month
w weeks will be (3w/13)th of a month
P(x) = 5 * 2^x OR
P(w) = 5 * 2^(3w/13)

Answer 18

For the last part,
P = 5 * 2^x
To convert an exponential function to a logarithmic function, we take log (to the base 10) on both sides of the equation:
log(P) = log(5*2^x) = log(5) + log(2^x) = log(5) + x * log(2)
log(P) = log(2) * x + log(5)

If you want you can substitute decimal values of log(2) and log(5) or you can just leave it the way it is.

Answer 19

Ok, I understand that but could it be done the way I was thinking?

Answer 20

Thank you for the last part.

Answer 21

For the middle part, they actually want you to convert the function so that instead of population being a function of x (the month) they want the population to be a function of w (the week). So if I want to know what the population is in week 5, I should be able to put w = 5 in the formula and calculate P.

The way you are doing, you are calculating the population after 1 year then dividing that by 52. That will be the averaging a years population growth equally among 52 weeks. That would assume a constant rate of growth week after week whereas this is an exponential growth.

Answer 22

If the question had been: Find the AVERAGE rate of growth per week (averaged over a one year period) then your method is correct.

Answer 23

Just noticed they want the RATE of growth, not just the population growth as a function of week!

The general exponential growth function is: y = a * (1 + r)^x, where a is the initial value and r is the growth rate.

P(w) = 5 * 2^(3w/13)
P(w) = 5 * (2^(3/13))^w
P(w) = 5 * 1.17346^w
P(w) = 5 * (1 + 0.17346)^w
0.17346 or 17.346% is the growth rate.

Answer 24

@ranga ... thank you so much... again... just got back to my computer and saw the rest of your post.

Answer 25

You are welcome.