What is the decay rate in the function A(t)=0.8(0.65)^t
To find the decay rate in the function A(t) = 0.8(0.65)^t, we need to first understand the mathematical form of the function. The function represents exponential decay, where "t" represents time and "A(t)" represents the amount or value of the decaying quantity at time "t".
In this case, the general form of an exponential decay function is given as A(t) = A₀ * e^(kt), where A₀ is the initial amount, "e" is the base of the natural logarithm (approximately 2.71828), "k" represents the decay rate, and "t" is time.
Comparing the given function A(t) = 0.8(0.65)^t with the general form, we can see that the initial amount A₀ is 0.8. However, instead of using "e", this function uses the base of 0.65.
To determine the decay rate, we can equate the given function to the general form of exponential decay and solve for "k".
0.8(0.65)^t = A₀ * e^(kt)
Since we know A₀ = 0.8, we can substitute it in:
0.8(0.65)^t = 0.8 * e^(kt)
Dividing both sides by 0.8:
(0.65)^t = e^(kt)
Now, to isolate the decay rate "k", we need to take the natural logarithm (ln) on both sides of the equation:
ln[(0.65)^t] = ln[e^(kt)]
Applying the logarithmic property ln(a^b) = b * ln(a):
t * ln(0.65) = kt * ln(e)
Since ln(e) = 1, the equation simplifies to:
t * ln(0.65) = kt
To solve for "k", we divide both sides by t and ln(0.65):
k = (t * ln(0.65)) / t
Simplifying:
k = ln(0.65)
Using a calculator, we can find that ln(0.65) ≈ -0.43078.
Therefore, the decay rate in the function A(t) = 0.8(0.65)^t is approximately -0.43078.