Grade 12 Logarithmic Math Problem Help!!!?
The population of a swarm of locusts can multiply tenfold in two weeks.
a) if there are 3000 locusts now, explain why an equation that would model the population over n weeks is given by An=(3000)(10^n/2)
b) how many locusts will there be after 7 weeks?
c) how long will it take for there to be 1 billion locusts?
Best answer will be given!!
Comments
A(sub n) = A(sub 0) * 10^(n / 2),
where A(sub n) is the amount at clock says time "n", and A(sub 0) is the amount at time "0" which is 3000 locusts, and 10^ means to 10 fold increase the population exponentially, the "n" is the number of weeks, and 2 is the the time that it takes the population to increase 10 fold which is "2 weeks".
This very much resembles the equations which model half lives of nuclear decaying material, the one that doesn't use euler's number.
b) How many locusts will there be after 7 weeks:
A(sub n is 7) = 3000 * 10^(7 / 2) ... just substitute "n" for 7:
= 9486832.98051 ... just use a calculator. I used Google's calculator.
c) How long will it take for there to be A(sub n is 1 billion locusts)? So we are looking for "n weeks". Just solve for n:
A(sub n) = A(sub 0) * 10^(n / 2)
Divide by A(sub 0) to cancel it's multiplication:
A(sub n) / A(sub 0) = 10^(n / 2),
and then calculate the logarithm (base 10 specifically since the base of the power is 10).
showing the operation:
log[A(sub n) / A(sub 0)] = log[10^(n / 2)]
log(A sub n) / A(sub 0)] = (n /2) * log(10) ... this is a property of logarithms pulling the exponent out of it.
And log base 10 of 10 is simply 1.
log(A sub n) / A(sub 0)] = (n / 2)
Now must multiply both sides of the equation by 2:
log[A(sub n) / (A sub 0)] * 2 = n
and we just input the known variables:
log[(10^6) / 3000] * 2 = n
about 5.045757 weeks = n
So in about 5 weeks there will be 1 billion locusts (unless they run out of resources which they will), though this model has no plateau.
Now we check our work:
A(sub n is 5.045757) = 3000 * 10^(5.045757 / 2)
= 999,999.435221 locusts
which is a 1 million locusts with a little round off error.