math  algebra
posted by Alexis on .
f(t) is an exponential function with
f(0) = 200 and f(3) = 2000.
a) Use the form f(t) = Pe^rt to find the equation of this exponential function.
b) In the form f(x) = Pa^x, a is the growth factor. find the growth factor.
c)What input value gives an output value of 5,000?

F(t) = Pe^rt
a. F(0) = P*e^0 = 200,
P*e^0 = 200,
P*1 = 200,
P = 200.
F(3) = 200*e^3r = 2000,
200e^3r = 2000,
Divide both sides by 200:
e^3r = 10,
Take Ln of both sides:
3r*Lne = Ln10,
3r = Ln10 / Lne = 2.30259 / 1 = 2.3026,
r = 0.7675.
Eq: F(t) = 200e^(0.7675t).
b. F(x) = 200a^x.
F(3) = 200a^3 = 2000,
200a^3 = 2000,
a^3 = 10,
a = 2.15443.
Eq: F(x) = 200(2.15443)^x
c. F(x) = 200a^x = 5000,
200a^x = 5000,
a^x = 25,
Take Log of botn sides:
x*Loga = Log25,
X = Log25 / Loga=Log25 / Log(2.15443),
X = 4.194.