SOLUTION: Some biologists model the number of species S in a fixed area A (such as an island) by the species-area relationship
log(S) = log(c) + k log(A)
where c and k are positive constan
Algebra.Com
Question 1170843: Some biologists model the number of species S in a fixed area A (such as an island) by the species-area relationship
log(S) = log(c) + k log(A)
where c and k are positive constants that depend on the type of species and habitat.
(a) Solve the equation for S.
S =
(b) Using part (a), if k = 2 and the area is doubled, then by what magnitude is the number of species increased?
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!
Part (a)
We'll use these log rules
log(A)+log(B) = log(A*B)
B*log(A) = log(A^B)
which we'll refer to as rule 1 and rule 2 respectively.
log(S) = log(c) + k*log(A)
log(S) = log(c) + log(A^k) .... use rule 2
log(S) = log(c*A^k) .... use rule 1
Since the log on the left equals the log on the right, this means the arguments (ie the stuff inside) for each log function must be equal.
In other words, log(A) = log(B) implies that A = B.
The equation
log(S) = log(c*A^k)
turns into
S = c*A^k
Answer: S = c*A^k
==========================================================================
Part (b)
The area is doubled. So A turns into 2A.
We'll replace A with 2A like so
S = c*A^k
S = c*(2A)^k
Now plug in k = 2
S = c*(2A)^k
S = c*(2A)^2
S = c*(4A^2)
Plug k = 2 into the original equation
S = c*A^k
S = c*A^2
Divide the two results
[ c*(4A^2) ] / [ c*A^2 ]
(4A^2)/(A^2)
4
The 'c's cancel and the 'A^2's cancel.
For k = 2, if we double the area from A to 2A, then the number of species has increased by a factor of 4. It turns out this works for any value of k as well.
Answer: 4
RELATED QUESTIONS
I would appreciate some input on this. My solutions are way off.
The function f(x) =... (answered by tutor_paul)
The population growth of an animal species is described by F(t)=500+90 log3 (2t+1) where... (answered by oscargut)
can someone help me with this problem:
The population growth of an animal species is... (answered by alicealc)
The population growth of an animal species is described by the equation:
F(t)=700+50... (answered by josmiceli)
A species of animal is discovered on an island. Suppose that the population size P(t) of... (answered by stanbon)
The Galapagos Islands, lying 600 miles west of the Ecuador, are famed for their... (answered by solver91311)
You can use the equation N = k log A to estimate the number of species N that live in a re (answered by rothauserc)
Solve the problem.
The population growth of an animal species is described by
F(t) = (answered by nerdybill)
Word problem
The population growth of an animal species is described by
F(t) = 700 + (answered by RAY100)