Answer:
The treatment decays by half each week.
Step-by-step explanation:
The decay function is given by:
Here,
<em>y</em> = final value
<em>a</em> = initial value
<em>r</em> = decay rate
<em>t </em>= time
The decay function for one medications to treat dogs for fleas is:
Here the decay rate is:
The treatment decays by half each week.
Answers:
a = -6/37
b = -1/37
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Explanation:
Let's start things off by computing the derivatives we'll need
Apply substitution to get
I've factored things in such a way that we have something in the form Msin(x) + Ncos(x), where M and N are coefficients based on the constants a,b.
The right hand side is simply sin(x). So we want that cos(x) term to go away. To do so, we need the coefficient (a-6b) in front of that cosine to be zero
a-6b = 0
a = 6b
At the same time, we want the (-6a-b)sin(x) term to have its coefficient be 1. That way we simplify the left hand side to sin(x)
-6a -b = 1
-6(6b) - b = 1 .... plug in a = 6b
-36b - b = 1
-37b = 1
b = -1/37
Use this to find 'a'
a = 6b
a = 6(-1/37)
a = -6/37
Base case: For , the left side is 2 and the right is
, so the base case holds.
Induction hypothesis: Assume the statement is true for , that is
We want to show that this implies truth for , that
The first terms on the left reduce according to the assumption above, and we can simplify the
-th term a bit:
so the statement is true for all .