Question

the plot in the tb - plane shows the brewing time, b, defined as the time it takes for a tea packet to reach a caffeine concentration of 100 milligrams per liter with water at a temperature of t degrees celsius. which of the following equations best models the relationship between b and t? choose 1 answer. b = 9 - 0.66^t b = 9 - 0.06^t b = 25 - 0.66^t b = 25 - 0.06^t

Explanation:

Step1: Analyze the trend of the data points

The data points show a decreasing - exponential relationship between \(B\) (brewing time) and \(T\) (temperature). The general form of an exponential decay function is \(y = a\cdot b^{x}\), where \(0 < b<1\). When \(x = 0\), \(y=a\).

Step2: Estimate the \(y\) - intercept

When \(T = 0\), from the graph, \(B\) is approximately \(9\). So, the value of \(a\) in the function \(B=a\cdot b^{T}\) is around \(9\).

Step3: Analyze the decay factor

As \(T\) increases, \(B\) decreases. We need to find a value of \(b\) such that the function fits the data. Since the decay is not too rapid, we can eliminate options with very small \(b\) values.

Step4: Check the options

For an exponential - decay function \(B=a\cdot b^{T}\), when \(a = 9\) and considering the decay rate of the data points, the function \(B = 9\cdot0.96^{T}\) is a reasonable fit. The function \(B = 9\cdot0.66^{T}\) would decay too rapidly, and functions with \(a = 25\) are not consistent with the \(y\) - intercept of the graph.

Answer:

B. \(B = 9\cdot0.96^{T}\)