All exponential functions can be written in many forms. Write the function f, of, t, equals, 8000, left bracket, 1, point, 4, right bracket, start superscript, start fraction, t, divided by, 20, end fraction, end superscriptf(t)=8000(1.4)

Question

20
t
​

in the form f, of, t, equals, a, e, start superscript, k, t, end superscriptf(t)=ae
kt
. Round all coefficients to four decimal places.

Answer 1

To write the function in the form f(t) = a * e^(kt), we need to express 1.4 as e^k, where k is a constant.

Let's equate the two forms of the function:

8000(1.4)^t/20 = a * e^(kt)

First, let's simplify the expression (1.4)^t/20:

(1.4)^t/20 = e^(kt)

Taking the natural logarithm of both sides:

ln((1.4)^t/20) = ln(e^(kt))

Using the logarithmic property ln(a^b) = b * ln(a):

t * ln(1.4) - ln(20) = kt * ln(e)

Since ln(e) = 1, the equation becomes:

t * ln(1.4) - ln(20) = kt

Comparing this equation with f(t) = a * e^(kt), we can determine the values of a and k:

a = ln(1.4) ≈ 0.3365 (rounded to four decimal places)
k = ln(20) ≈ 2.9957 (rounded to four decimal places)

Therefore, the function can be written as:

f(t) = 0.3365 * e^(2.9957t) (rounded to four decimal places)