All exponential functions can be written in many forms. Write the function f, of, t, equals, 10000, left bracket, 1, point, 2, 5, right bracket, start superscript, start fraction, t, divided by, 2, end fraction, end superscriptf(t)=10000(1.25)
2
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.
To write the function in the desired form, we need to convert the base from 1.25 to e. We know that e is approximately 2.71828. Now let's find the value of a and k.
We have:
f(t) = 10000(1.25)^{(2t)}
To convert the base to e, we need to rewrite 1.25 as e^k. Let's solve for k:
e^k = 1.25
Taking the natural logarithm of both sides, we have:
k = ln(1.25)
Now we can write the function in the desired form:
f(t) = a*e^(kt)
To find the value of a, we can substitute a known value of f(t) and solve for it. Let's use f(0) = 10000:
10000 = a*e^(k*0)
10000 = a*e^0
10000 = a
So, a = 10000.
Now we can write the function in the desired form with rounded coefficients:
f(t) = 10000 * e^(ln(1.25)*t)