Question

Please help. The drop downs are both decay.

Answer 1

(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.