Answer: An exponential decay function models a situation in which a value decreases over time at a rate that is proportional to its current value. The function has the form:
y = A*e^(-kt)
Where y is the final value, A is the initial value, e is the base of the natural logarithm (approximately 2.718), k is the decay factor, and t is the time.
In this case, the initial value is 42 and the decay factor is 0.7. So the exponential decay function that models the situation is:
y = 42 * e^(-0.7t)
To compare the average rates of change over the given intervals, we need to find the derivative of the function, which represents the instantaneous rate of change of the function at any given point in time.
The derivative of this function is:
dy/dt = -0.742e^(-0.7t)
The negative sign indicates that the value is decreasing over time. The absolute value of the derivative will be the rate of decay.
So, the average rate of change over the given intervals is -0.7*42 which is -29.4
The rate of decay is always -29.4 for this function.
Explanation: