I have 2 sets of coordinates (1, 410) and (12, 788) and need to calculate the rate of change over time using y=ae^bt. Since I don't have a coordinate at 0, I can't figure out how to calculate my starting value, or a, to solve this.
y=ae^bt
a) 410=ae^(b*1)
b) 788=ae^(12b)
divide equation a) by b)
410/788=e(b-12b)
take ln of each side
-11b=ln(420/788)
b= .057
Now , solve for a.
410=ae^(.057*1)
a= 410/1.0587=387
y= 387e^(.057t)
To calculate the rate of change over time using the equation y = ae^bt, you need to first determine the values of a and b. However, since you do not have a coordinate at time t = 0, it becomes challenging to find the exact values of a and b.
To proceed, you can make an assumption about the initial value of y (y₀) and then calculate a and b accordingly. One common approach is to assume that the initial value of y (y₀) is equal to the y-coordinate of your first data point, which in this case is 410.
With this assumption, you can rewrite the equation as y = 410e^bt. Now, you need to find the value of b. To do this, you can use the second data point (12, 788).
Substituting x = 12 and y = 788 into the equation, you get 788 = 410e^(12b).
To isolate b, divide both sides of the equation by 410, which gives you:
788/410 = e^(12b).
Now, take the natural logarithm (ln) of both sides:
ln(788/410) = 12b.
Simplifying further, you can determine the value of b by dividing the natural logarithm of 788/410 by 12:
b = ln(788/410) / 12.
Once you have calculated the value of b, you can substitute it back into the equation (y = 410e^bt) along with the assumed initial value of y (y₀ = 410) to find the value of a:
a = y₀ / e^(bt_0).
Using the given y₀ = 410 and b calculated from the previous step, you can compute a:
a = 410 / e^(bt_0).
Now you have both a and b, allowing you to determine the rate of change over time using the exponential equation y = ae^bt for any given time (t).