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.
a) To find the equation of the exponential function, we'll use the given form f(t) = Pe^rt and the information given: f(0) = 200 and f(3) = 2000.
We know that f(0) = 200, so we can substitute this into the equation: 200 = Pe^r(0). Since any number raised to the power of 0 is always 1, the equation simplifies to: 200 = P.
Now, we can find the value of P. We know that f(3) = 2000, so we substitute this into the equation: 2000 = 200e^r(3). We can divide both sides by 200 to isolate e^r(3): 2000/200 = e^r(3). Simplifying further, we have: 10 = e^r(3).
To solve for r, we can take the natural logarithm (ln) of both sides: ln(10) = ln(e^r(3)). Using the property of logarithms, ln(e^x) = x, the equation becomes: ln(10) = r(3).
So, the equation of the exponential function is f(t) = 200e^(ln(10)/3)t.
b) In the form f(x) = Pa^x, the growth factor is represented by 'a'. To find 'a', we can rewrite the equation f(t) = 200e^(ln(10)/3)t in the form f(x) = Pa^x.
Comparing both equations, we can see that 'a' is equal to e^(ln(10)/3). Using the property that e^(ln(x)) = x, the expression simplifies to: a = 10^(1/3).
Therefore, the growth factor 'a' is equal to 10^(1/3).
c) To find the input value that gives an output value of 5,000, we can use the equation of the exponential function: f(t) = 200e^(ln(10)/3)t.
Substitute '5000' for 'f(t)' and solve the equation for 't':
5000 = 200e^(ln(10)/3)t
Divide both sides by 200:
25 = e^(ln(10)/3)t
To isolate 't', we need to take the natural logarithm (ln) of both sides:
ln(25) = ln(e^(ln(10)/3)t)
Using the property of logarithms, ln(e^x) = x, the equation simplifies to:
ln(25) = (ln(10)/3)t
Solve for 't' by multiplying both sides by 3/ln(10):
t = ln(25) / (ln(10)/3)
Using a calculator, we can find that t ≈ 2.535.
Therefore, the input value that gives an output value of 5,000 is approximately 2.535.