if a virus is found in a body and after 1h there are 80 bacteria in the individual. and after 100mins (1h&40mins) there are 320 bacteria.
find the equation of N(t)=a(b^t) where a is the initial value and b is the growth rate and t is time in hours.
ive tried this many times and i just cant figure it out. ive asked my friends and they dont know either
please help
?
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Plugging in the given numbers, N(1) = a(b^1) = 80 => a*b = 80.
and 100mins = ~1.67 hrs (really 1&2/3 hrs) , so N(1.67) = a(b^1.67) = 320.
If I solve the first equation for a, I get a = 80/b. Substituting that into equation 2 gives me
(80/b)(b^1.67) = 320.
move the 80/b to the other side so that the b^1.67 is by itself. => b^1.67 = 320(b/80) =>b^1.67 = 4b
--------This is where it gets tricky-----------
(When in doubt take the log of both sides)
Here we're going to need logarithms, and you will want to use log base b to cancel as much as you can. to denote log base b I will write log{b}.
So, log{b}(b^1.67) = log{b}(4b) => 1.67*log{b}(b) = log{b}(4) + log{b}(b) => 1.67 = log{b}4 + 1
=> log{b}(4) = 2/3 (Note that the real decimal is not 0.67 it is a repeating decimal that can be rewritten as 2/3)
=>log{b}(4) = 2/3 => b^(2/3) = 4 => b = 4^(3/2) => b = 8
Now that we've solved for b, we can plug that into the first equation (a*b=80) to get a.
a(8) = 80 => a=10.
This we can plug into the given formula N(t) = 10(8^t).
---------DONE---------
To check, plug your given information back into your solution.
N(1) = 10(8^1) = 80 (good)
N(1.67) = 10(8^1.67) = 320 (good)
?
Starting time = 1 hour = 60 mins
Starting value = 80
Next time = 1 hour 40 mins = 100 mins
next value = 320
Growth = 320-80/100-60 = 6 per minute.
So growth rate per hour = 6*60 = 360 per hour.
N(t) = 80(360^t)
Ray
n=ab^t
80=a*b^60 and 320=a*b^100
ln80=lna+60lnb and ln320=lna+100lnb
solve this system for lna and lnb
subtract eq1 from eq2: ln320-ln80= (100-60)lnb or ln4=40lnb and lnb=(ln4)/40
5*eq1: 5ln80=5lna + 300lnb
3*eq2: 3ln320= 3lna + 300lnb
subtract them: 5ln80-3ln320= 2lna; ln[80^5/320^3]=2lna; lna=(1/2)ln(80^5/320^3)
Now since lna=(1/2)ln(80^5/320^3), a=sqrt(80^5/320^3) or 10
And lnb=(ln4)/40 and b=4^(1/40) or 1.035265
So we have n= 10*1.035265^t
This answer can be verified by substituting in values of 60 and 100 for t. It checks out.