Answer: The decay factor is approximately 0.9984.
Explanation:
In an exponential function of the form A = Ab^x, the decay factor b is the constant factor by which the function decays or grows over each unit of time or input change. It is always a value between 0 and 1 in the case of exponential decay.
To find the decay factor in the given exponential function A = 0.46*74.4^x, we need to rewrite the function in the form A = Ab^x. We can do this by dividing both sides of the equation by the initial value of A:
A/A = (0.46*74.4^x)/A
Simplifying this expression gives us:
1 = 0.46*(74.4^x)/A
Dividing both sides by 0.46 and rearranging, we get:
(74.4^x)/A = 1/0.46
Simplifying further gives us:
(74.4^x)/A = 2.1739
Now, we can solve for the decay factor b by taking the logarithm of both sides of the equation:
log(b) = log(74.4)/ln(10)
Simplifying this expression using a calculator gives us:
log(b) = -0.0016
Taking the antilogarithm of both sides, we get:
b = 10^(-0.0016)
Approximating this value to four decimal places gives us:
b ≈ 0.9984
Therefore, the decay factor for the given exponential function is approximately 0.9984.