Find the initial value a, growth/decay factor b, and growth/decay rate r for the following exponential function:
Q(t)=0.0036(2.58)^−4t
a) The initial value is a=
(b) The growth factor is b=
(c) The growth rate is r=
To find the initial value (a), growth/decay factor (b), and growth/decay rate (r) for the exponential function Q(t) = 0.0036(2.58)^-4t, you can break down the equation and compare it to the standard form of an exponential function, which is:
Q(t) = a * b^rt
In this case, we have:
- The initial value (a) is the value of Q(t) when t = 0.
- The growth/decay factor (b) is the base of the exponential function.
- The growth/decay rate (r) indicates whether the function is growing (r > 0) or decaying (r < 0) and by how much.
Let's find these values for the given exponential function:
(a) Initial value (a):
To find the initial value (a), substitute t = 0 into the equation:
Q(0) = 0.0036(2.58)^(-4 * 0)
Q(0) = 0.0036(2.58)^0
Q(0) = 0.0036 * 1
Q(0) = 0.0036
Therefore, the initial value (a) is 0.0036.
(b) Growth/decay factor (b):
The growth/decay factor (b) is the base of the exponential function. In this case, the base is 2.58.
Therefore, the growth/decay factor (b) is 2.58.
(c) Growth/decay rate (r):
The growth/decay rate (r) is calculated by multiplying the exponent of the base by -4.
Therefore, the growth/decay rate (r) is -4.
To summarize:
(a) The initial value is a = 0.0036.
(b) The growth factor is b = 2.58.
(c) The growth rate is r = -4.
To find the initial value (a), growth/decay factor (b), and growth/decay rate (r) for the given exponential function Q(t) = 0.0036(2.58)^(-4t), we can use the general form of an exponential function:
Q(t) = a * b^(r*t)
Comparing this form with the given function:
Q(t) = 0.0036(2.58)^(-4t)
We can see that:
a = 0.0036
b = 2.58
and r = -4.
So, the answers are:
(a) The initial value is a = 0.0036
(b) The growth/decay factor is b = 2.58
(c) The growth/decay rate is r = -4.
(a) recall that 2.58^0 = 1
(bc) 2.58^-4 = 0.0225 = 2.25%