Advertising Revenue. The amount of advertising revenue in...

Advertising Revenue. The amount of advertising revenue in U.S. newspapers has declined continually since $2006 .$ In $2006,$ the advertising revenue was $\$ 49.3$ billion, and by 2013 that amount had decreased to $\$ 20.7$ billion (Source: Newspaper Association of America). Assuming that the amount of newspaper advertising revenue decreased according to the exponential decay model: a) Find the value of $k$, and write an exponential function that describes the advertising revenue after time $t,$ in years, where $t$ is the number of years after 2006 b) Estimate the advertising revenue in 2008 and in 2012 c) At this decay rate, when will the advertising revenue be $\$ 16$ billion?

Advertising Revenue. The amount of advertising revenue in U.S. newspapers has declined continually since $2006 .$ In $2006,$ the advertising revenue was $\$ 49.3$ billion, and by 2013 that amount had decreased to $\$ 20.7$ billion (Source: Newspaper Association of America). Assuming that the amount of newspaper advertising revenue decreased according to the exponential decay model:
a) Find the value of $k$, and write an exponential function that describes the advertising revenue after time $t,$ in years, where $t$ is the number of years after 2006
b) Estimate the advertising revenue in 2008 and in 2012
c) At this decay rate, when will the advertising revenue be $\$ 16$ billion?

Key Concepts

Function Modeling and Prediction

Once an exponential function is established using the initial value and decay constant, it can be used to predict future or past values of the diminishing quantity. This involves substituting specific time values into the model to estimate the corresponding outputs.

Exponential Decay Model

This concept describes processes in which a quantity decreases at a rate proportional to its current amount. The model is generally represented as A(t) = A?e^(–kt), where A? is the initial value, k is the decay constant, and t is time. It is useful for describing phenomena like radioactive decay, cooling, and financial depreciation.

Decay Constant (k) Calculation

The decay constant, k, quantifies the rate at which the exponential decay occurs. It is determined by using two data points from the process and solving the equation A(t) = A?e^(–kt) for k. The process typically involves taking the natural logarithm to isolate and calculate k.

Solving Exponential Equations

In many practical problems, a particular value of the decaying quantity is given, and the task is to determine the time at which this value is reached. This is done by setting up the exponential equation with the given value and solving for t, usually through the use of logarithms.

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