The population of termites and spiders in a certain house are growing exponentially. the house contains 100 termites the day you move in. After 4 days, the house contains 200 termites. Three days after moving in, there are two times as many termites as spiders. Eight days after moving in, there were four times as many termites as spiders/ Hows long in days does it take the population of spiders to triple?
What I done so far.
TERMITES
y(not)=100
t=4
y=200
200=100b^4
b=2^(1/4)
T(t)=100(1.189207)^t
for the spiders.S(t).. i tried plugging t=3 and t=8and divided by two and four to the T(x) equation.... and for two numbers 84.089 and 99.99..... but I think Im doing it wrong
or is it
T(t) = 2(S(t))?
I agree with your first part: b=2^(1/4) = 1.189207.
Now, after 3 days, according to the question, we must have 84.089 spiders - wonder what .089 of a spider looks like? - and after 8 days, we have 100.
If S0 is the number of spiders on day zero, and s is the multiplication rate of spiders, then
S0 * s^3 = 84.089
S0 * s^8 = 100
From which we can quickly get s^5.
And then we get s.
And from there, there's not much left to do, I think.
I made a mistake. after eight days , its 199.99 spiders.
I still don't understand how would we get s?
You didn't make a mistake:
"Eight days after moving in, there were four times as many termites as spiders"
But 100 * (2^.25)^8 = 400 (termites)
So if there were 400 termites, there were 100 spiders.
So how do we get s?
S0 * s^3 = 84.089
S0 * s^8 = 100
(S0 * s^8) / (S0 * s^3)
= s^5
= 100 / 84.089
s = (100 / 84.089)^(1/5)
Ohh all i needed to do was divide! heh thanks a lot!
You're welcome!
you guys never gave an answer..
To find the population of spiders over time, we can start by using the information given. Let's assume the initial population of spiders is S(0). We know that after 3 days, the termites are twice as many as the spiders. So, after 3 days, the termite population is 2*S(3).
We also know that after 8 days, the termites are four times as many as the spiders. So after 8 days, the termite population is 4*S(8).
Now, let's write the exponential growth equations for both termites and spiders:
Termites: T(t) = T(0) * b^t
Spiders: S(t) = S(0) * c^t
Using the information given, we can find the values of b and c.
From the data given, we have:
T(0) = 100 (initial population of termites)
T(3) = 200 (population of termites after 3 days)
T(8) = 4*S(8) (population of termites after 8 days, 4 times the spider population)
Using the equation for termites: T(t) = T(0) * b^t
Substituting the values:
200 = 100 * b^3 (equation 1)
4S(8) = 100 * b^8 (equation 2)
Now, let's find the value of b by solving equation 1:
200 = 100 * b^3
b^3 = 200/100
b^3 = 2
b = 2^(1/3) ≈ 1.26
Now, let's find the value of c by solving equation 2:
4S(8) = 100 * b^8
S(8) = (100 * b^8) / 4
S(8) = 25 * b^8
Now, to find the population of spiders after 8 days, we can use the equation S(8) = S(0) * c^8:
25 * b^8 = S(0) * c^8
Since we don't know the value of S(0), we can assume it to be 1. Thus:
25 * b^8 = c^8
Taking the 8th root of both sides:
c = (25 * b^8)^(1/8)
Now, to find the population of spiders after 4 days, we can use the equation S(4) = S(0) * c^4:
S(4) = S(0) * c^4
Assuming S(0) = 1:
S(4) = c^4
To find the population of spiders after 4 days, we need to find the value of c and then calculate c^4.
Substituting the calculated value of b in the equation for c:
c = (25 * b^8)^(1/8)
c = (25 * (2^(1/3))^8)^(1/8)
c ≈ 1.915
Now, we can find the population of spiders after 4 days:
S(4) ≈ 1.915^4
S(4) ≈ 14.78
Therefore, the population of spiders after 4 days is approximately 14.78.
To answer the question of how long it takes for the spider population to triple, we need to find t such that:
3 * S(0) = S(t)
Assuming S(0) = 1:
3 * 1 ≈ S(t)
3 ≈ S(t)
Therefore, the population of spiders needs to reach approximately 3 for the spider population to triple.
To find the value of t, we can solve the exponential growth equation for spiders:
S(t) = S(0) * c^t
Substituting the values:
3 = 1 * c^t
Taking the logarithm of both sides:
log(3) = log(c^t)
log(3) = t * log(c)
Dividing both sides by log(c):
t = log(3) / log(c)
Using the value of c ≈ 1.915 and solving for t:
t ≈ log(3) / log(1.915)
t ≈ 1.585 / 0.280
t ≈ 5.662
Therefore, it takes approximately 5.662 days for the population of spiders to triple in size.