Question

Use the following information. From 1894 to 1903 the number of miles of cable car track decreased by about \(10 \%\) per year. There were 302 miles of track in 1894 . Write an exponential decay model showing the number of miles \(M\) of cable car track left in year \(t.\)

Short answer

The exponential decay model showing the number of miles \(M\) of cable car track left in year \(t\) is expressed by the equation \(M(t) = 302 * (0.90)^t\).

Step-by-step solution

Understand the Exercise

The exercise provides that in 1894 there were 302 miles of track and this decreased by about 10 \% per year. This indicates that the amount of track is reducing in an exponential way. Therefore, an exponential decay model should be created.

Identify the Decay Rate and Initial Value

The decay rate, \(r\), is the rate at which the number of miles is decreasing each year, in this case, 10\% or 0.10. The initial value, \(M_0\), is the number of miles in 1894, which we know to be 302 miles.

Writing the Exponential Decay Equation

Substituting the values of \(M_0\) and \(r\) into the equation for exponential decay, we get the equation \(M(t) = 302 * (1 - 0.10)^t\). This is the model showing the number of miles \(M\) of cable car track left in year \(t\).