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Please help. The drop downs are both decay.

Please help. The drop downs are both decay.-example-1

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(a) The exponential function
\(y = 400e^(-0.07t)\) can be rewritten as
\(y = 400(0.9349)^t\), where
\(b \approx 0.9349\). (b) The annual decay rate is approximately 6.51% per year (rounded to the nearest 0.01%). (c) The continuous decay rate is approximately -6.72% per year (rounded to the nearest 0.01%).

(a) To rewrite the exponential function
\(y = 400e^(-0.07t)\) in the form
\(y = ab^t\), recognize that
\(e^(-0.07) = b\). Therefore,


\[y = 400e^(-0.07t) = 400(b^t) \]

(b) Calculate the value of b using
\(e^(-0.07)\):


\[b = e^(-0.07) \approx 0.9349\]

The exponential function is decaying because the base (b) is between 0 and 1.

(c) For the annual growth/decay rate r, use the formula r = |1 - b| and multiply by 100 to express as a percentage:

Annual decay rate =
|1 - 0.9349| * 100 \approx 6.51\%

For the continuous growth/decay rate, use the natural logarithm ln of b, multiplied by 100:


\text{Continuous decay rate} = \ln(0.9349) * 100 \approx -6.72\%

The negative sign indicates decay in both cases, as mentioned.

User Dgrijuela
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