write the exponential function y=40e^0.06t in the form y=ab^t.
find b accurate to 4 decimal palces.
if t is measured in years, give the % annual growth or decay rate and the continuous % growth or decay rate per year.
y = 40e^.06t = 40(e^.06)^t = 40*1.0618^t
annual growth rate: 6.18%
continuous rate: 6.00%
To write the exponential function y = 40e^(0.06t) in the form y = ab^t, we need to express the base 'e' as 'b' to a certain power.
First, let's understand the relationship between 'e' and 'b'. We know that e is Euler's number and it is approximately equal to 2.71828. The base 'b' we are looking for is related to 'e' as follows:
b = e^k, where k is a constant.
To find 'b', we need to find the value of 'k'. We know that e^(0.06t) is equivalent to (e^(0.06))^t, so we can rewrite the equation y = 40e^(0.06t) as follows:
y = 40(e^(0.06))^t
Comparing this equation with y = ab^t, we can see that a = 40 and b = (e^(0.06)).
To find 'b' accurate to 4 decimal places, we substitute the value of 'e' and 'k' into the equation:
b = (2.71828)^(0.06)
Calculating this gives: b ≈ 1.0618 (rounded to 4 decimal places).
Now, let's determine the % annual growth or decay rate and the continuous % growth or decay rate per year.
The % annual growth or decay rate is given by the formula: r = 100(b - 1).
Substituting the value of 'b', we have:
r = 100(1.0618 - 1) ≈ 6.18%.
The continuous % growth or decay rate per year is given by the formula: k = 100ln(b).
Substituting the value of 'b', we have:
k = 100ln(1.0618) ≈ 6.10%.
Therefore, the % annual growth or decay rate is approximately 6.18%, and the continuous % growth or decay rate per year is approximately 6.10%.