Final answer:
To find the exponential model, use the given amounts at 6 and 10 hours to set up equations and solve for the constants 'a' and 'b'. Then, use these values to predict the amount remaining after 12 hours, understanding that this is an exponential decay situation.
Step-by-step explanation:
To find an exponential model y = abt that represents the decay of a substance, we can use two given data points: (6, 270) and (10, 242). The variable 'a' represents the initial amount of the substance, 'b' is the base of the exponential function which represents the decay rate per unit time, and 't' is the time in hours. Since we have the data for six hours and ten hours into the experiment, we can set up two equations to solve for 'a' and 'b'.
First equation for the 6-hour mark: 270 = a * b6
Second equation for the 10-hour mark: 242 = a * b10
To solve for 'a' and 'b', we can divide the second equation by the first equation in order to eliminate 'a':
242/270 = (a * b10) / (a * b6)
This gives us b10 / b6 = b10 - 6 = b4, so:
242/270 = b4
Now we find the fourth root of (242/270) to find 'b':
(242/270)1/4 = b
Next, using the value of 'b', we can solve for 'a' from the first equation:
270 = a * b6
Finally, using the calculated values of 'a' and 'b', we can predict the remaining amount of the substance after 12 hours:
amount remaining after 12 hours = a * b12
Note that this does not involve using a specific half-life, but rather the given data points for the exponential decay problem.