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?
(a) If f(t) = Pe^(rt)
200 = P^0 = P (Therefore P = 200)
2000 = 200 e^(3r)
e^3r = 10
3r = ln10 = 2.303
r = 0.76753
The equation is
f(t) = 200*e^(0.76753 t)
(b) 2000 = 200*a^3
a^3 = 10
a = 2.15443
y = 200*(2.15443)^x
(c) Solve 5000 = 200*2.15443^x
25 = 2.15443^x
x = ln25/ln2.15443 = 4.1938
To find the equation of the exponential function, we can use the given points (0, 200) and (3, 2000) to solve for the parameters P and r.
a) Start by substituting the values f(0) = 200 and f(3) = 2000 into the exponential function formula f(t) = Pe^rt:
f(0) = Pe^(0*r) = P
f(3) = Pe^(3*r)
Substituting the given values:
200 = P
2000 = Pe^(3*r)
Now we have a system of two equations:
Equation 1: P = 200
Equation 2: 2000 = 200e^(3r)
Divide equation 2 by equation 1 to eliminate P:
(2000/200) = (200/200)e^(3r)
10 = e^(3r)
Take the natural logarithm (ln) of both sides to solve for r:
ln(10) = ln(e^(3r))
ln(10) = 3r
Divide both sides by 3 to get r:
r = ln(10)/3
Now that we have the value of r, we can substitute it back into equation 1 to find P:
P = 200
Therefore, the equation of the exponential function is:
f(t) = 200e^(ln(10)/3 * t)
b) The growth factor (a) is related to the parameter r by the equation a = e^r. Using the value of r we found in part a), we can calculate the growth factor:
a = e^(ln(10)/3)
a = 10^(1/3)
Therefore, the growth factor is 10^(1/3).
c) To calculate the input value given the output value of 5000, we can use the equation of the exponential function found in part a):
5000 = 200e^(ln(10)/3 * t)
Rearrange the equation and solve for t:
e^(ln(10)/3 * t) = 5000/200
e^(ln(10)/3 * t) = 25
Take the natural logarithm (ln) of both sides to get rid of the exponential:
ln(e^(ln(10)/3 * t)) = ln(25)
(ln(10)/3 * t) = ln(25)
Solve for t:
t = ln(25)/(ln(10)/3)
Therefore, the input value that gives an output value of 5000 is t = ln(25)/(ln(10)/3).