Question

Rewrite the function to determine whether it represents exponential growth or exponential decay. Identify the percent rate of change. Round numbers to the nearest hundredth, if necessary. y=(1.06)8t In the form y=a(1+r)t, the function is y≈ . To the nearest percent, the rate of change is a %.

Answer 1

The given function represents exponential growth with a rate of change of 6%. The function already fits the form y = a(1 + r)^t with a = 1 and r = 0.06.

Step-by-step explanation:

To rewrite the function y = (1.06)^8t in the form y = a(1 + r)^t, we can identify a as the initial amount and r as the rate of change. In this case, the initial amount a is implied to be 1 (since it is not explicitly multiplied by the base), and the base 1.06 represents the growth factor. Seeing as the base is greater than 1, this function represents exponential growth.

The percent rate of change can be found by subtracting 1 from the base and then converting it to a percentage. Therefore, the rate r is 1.06 - 1, which equals 0.06 or 6% when expressed as a percentage. Thus, the function in the requested form is y = (1 + 0.06)^8t, or simplifying the exponent, y = (1.06)^8t.

To the nearest percent, the rate of change is 6%.

Answer 2

y ≈1*(1+0.59)^t. The rate of change is = 59%

Step-by-step explanation:

We have t mulitplied by 8 in the expression of y. We can write that power with a power of powers, using the property

`a^(bc) = (a^b)^c = (a^c)^b`

Therefore,

`a^(8t) = (a^8)^t`

If we replace a with 1.06, we obtain

`y = 1.06^(8t) = (1.06^8)^t \approx 1.59^t = 1*(1+0.59)^t`

Thus, y ≈1*(1+0.59)^t. The rate of change is ln(1.59) * 1.59^t. After 1 unit of time t, the rate of change is 0.59*100 = 59%