Question

Y=78(0.54)^t Identify the initial amount a and the rate of growth r (as a percent) of the exponential function. Evaluate the function when t=5. Round your answer to the nearest tenth.

Answer 1

Initial amount = 78

Rate of growth = -46%

Y(t=5) = 3.6

Explanation:

The general form of an exponential function is:

`Y(t) = Y_0*(1+r)^t`

The initial value (Y0) is the value of Y(t) when t = 0:

`Y(0) = 78*0.54^0=78`

Comparing the given function to the general form of the exponential function, the rate of growth 'r' is determined as:

`Y(t) = Y_0*(1+r)^t = 78*(0.54)^t\\78*(1+r)^t = 78*(0.54)^t\\1+r=0.54\\r=-0.46=-46\%`

At t=5, the value of Y(t) is:

`Y(5)=78(0.54)^5\\Y=3.6`

Answer 2

`y = 78(0.54)^t`

And for this case we want to identify the initial amount and for this case would be 78. And the growth rate can be obtained from the term inside of the parenthesis.

The general model is given by this model:

`y = a (1+r)^t`

And then
`1+r = 0.54` and
`r = 0.54 -1= -0.46`

And then we want to find the of the function at t=5 and we got:

`y(5) = 78(0.54)^5 = 3.58 \approx 3.6`

Explanation:

For this case we have the following function given:

`y = 78(0.54)^t`

And for this case we want to identify the initial amount and for this case would be 78. And the growth rate can be obtained from the term inside of the parenthesis.

The general model is given by this model:

`y = a (1+r)^t`

And then
`1+r = 0.54` and
`r = 0.54 -1= -0.46`

And then we want to find the of the function at t=5 and we got:

`y(5) = 78(0.54)^5 = 3.58 \approx 3.6`