4) The function f(x)=2(3)^x represents the growth of a dragonfly population every year in a remote swamp. Rose wants to manipulate the formula to an equivalent form that calculates seven times a year, not just once a year. Which function is correct for Rose’s purpose and what is the new growth rate?
A) f(x)=2(3)^x, growth rate 200%
B) f(x)=2(1.17)^7x, growth rate 17%
C) f(x)=2(3)^x,growth rate 2%
D) f(x)=2(1.17)^x,growth rate 2%
I do not know what to do with this question. I am very confused. I'm thinking it is B or D, but maybe D?
Please explain this to me!! :)
If I understand your question correctly, this is the same as finding the equivalent of an annual rate with new compounding frequencies
the way it stands, f(x) = 2(3)^x
or f(x) = 2( 1 + 2)^x to me represents a growth of 200% compounded annually
so you want
f(x) = 2(1 + r)^7x to be equivalent
2(3)^x = 2(1+r)^7x
3^x = (1+r)^7x
take log of both sides
x log3 = 7x log (1+r)
log 3 = 7log(1+r)
log3 = log( (1+r)^7)
3 = (1+r)^7
take 7th root of both sides
1.16993 = 1+r
r = .16993 or 16.993%
so f(x) = 2(1.16993)^(7x) should do it
which looks like B
I was going to check this:
e.g how much after 3 years
original:
f(3) = 2(3)3 = 54
new one:
f(3) = 2(1.16993)^21 = 53.9992 , not bad
According to B
f(3) = 2(1.17)^21 = 54.067 ,
so it is B
To calculate the growth of the dragonfly population seven times a year instead of once a year, we need to modify the exponential growth formula. Let's go through the options one by one.
A) f(x)=2(3)^x, growth rate 200%
This option does not account for increasing the growth frequency, so it is not correct.
B) f(x)=2(1.17)^7x, growth rate 17%
In this option, the original growth rate of 17% is compounded seven times a year. However, the base, 1.17, does not match the original base, which is 3. So, this option is not correct.
C) f(x)=2(3)^x, growth rate 2%
This option is the same as the original formula provided. It does not account for increasing the growth frequency, so it is not correct.
D) f(x)=2(1.17)^x, growth rate 2%
In this option, the original growth rate of 2% is compounded once a year. This option changes the base from 3 to 1.17, which corresponds to a 2% growth rate. Therefore, this option is correct.
So, the correct function for Rose's purpose is f(x) = 2(1.17)^x, with a new growth rate of 2%.
To solve this problem, we need to understand and manipulate exponential growth. The original function f(x)=2(3)^x represents the growth of the dragonfly population by a factor of 3 each year.
To calculate the growth seven times per year, we can divide the growth rate (3) by the number of times per year (7). This gives us a new growth rate of 3/7.
Now let's look at the answer choices:
A) f(x)=2(3)^x, growth rate 200% - This answer choice doesn't reflect the change to the growth rate that Rose wants, so it's incorrect.
B) f(x)=2(1.17)^7x, growth rate 17% - This answer choice does include the change to seven times a year, as we see the exponent 7x. However, the growth rate of 17% doesn't align with the modified growth rate of 3/7. So, it's incorrect.
C) f(x)=2(3)^x, growth rate 2% - This answer choice doesn't reflect the change to the growth rate that Rose wants, so it's incorrect.
D) f(x)=2(1.17)^x, growth rate 2% - This answer choice includes the change to the growth rate of 3/7, as we see the base 1.17. Additionally, the growth rate of 2% aligns with the new growth rate of 3/7. Therefore, this is the correct answer.
So, the correct function for Rose's purpose is f(x)=2(1.17)^x, with a new growth rate of 2%.