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?
mr.jagodnishvil
200 = Pe^0 = P
2000 = Pe^rt = 200e^3r
e^3r = 10
3r = ln 10
r = ln10/3 = 0.7675
f(x) = 200a^x = 200e^.7675x
a^x = e^.7675x
a = e^.7675 = 2.154
5000 = 2.154^x
ln 5000 = x ln 2.154
x = ln5000/ln2.154 = 11.1
To find the equation of the exponential function, we will use the given points (0, 200) and (3, 2000).
a) Using the formula f(t) = Pe^rt, we can substitute the values (0, 200) into the equation to find the value of P.
So, we get 200 = P * e^(r * 0).
Since anything raised to the power of 0 is 1, the equation simplifies to 200 = P * 1, which means P = 200.
Now we can use the second point (3, 2000) to find the growth rate r. Using the equation f(t) = Pe^rt again, we substitute the values (3, 2000) to get 2000 = 200 * e^(r * 3).
Dividing both sides of the equation by 200, we have 10 = e^(3r).
To solve for r, we take the natural logarithm (ln) on both sides of the equation:
ln(10) = ln(e^(3r)).
Using the property ln(e^x) = x, we get ln(10) = 3r.
Finally, we solve for r by dividing both sides of the equation by 3:
r = ln(10)/3.
Therefore, the equation of the exponential function is f(t) = 200e^(ln(10)/3 * t).
b) The growth factor is represented by a in the form f(x) = Pa^x. To find a, we need to rewrite the equation f(t) = 200e^(ln(10)/3 * t) in the form f(x) = Pa^x.
Using the property e^(ln(x)) = x, the equation simplifies to f(t) = 200(e^(ln(10)/3))^t.
So, the growth factor a is e^(ln(10)/3).
c) To find the input value that gives an output value of 5,000 (f(t) = 5000), we can substitute this value into the equation f(t) = 200e^(ln(10)/3 * t) and solve for t.
5000 = 200e^(ln(10)/3 * t).
Dividing both sides by 200, we have:
25 = e^(ln(10)/3 * t).
Taking the natural logarithm (ln) on both sides of the equation, we get:
ln(25) = ln(e^(ln(10)/3 * t)).
Using the property ln(e^x) = x, the equation simplifies to ln(25) = ln(10)/3 * t.
Solving for t, we divide both sides by ln(10)/3:
t = ln(25) / (ln(10)/3).
Evaluating this expression, we find the value of t, which gives an output of 5,000.