time(days) remaining amount of sucralose(mg) 0 40 2 20 4 10 6 5 8 2.5
formula is y = a * b^x
a is the initial value.
b is the base for the exponent.
x is the exponent.
from the table, you get 40 = 40 * b^0.
since b^0 is always equal to 1, you get 40 = 40 and b is still undefined.
from the table, you get 20 = 40 * b^2.
solve for b^2 to get:
b^2 = .5
solve for b to get:
b = .7071067812.
from the table, you get 10 = 40 * b^4.
solve for b^4 to get:
b^4 = .25
solve for b to get:
b = .25^(1/4) = .7071067812.
from the table, you get 5 = 40 * b^6.
solve for b^6 to get:
b^6 = 1/8
solve for b to get:
b = (1/8)^(1/6) = .7071067812.
this very much indicates that b = .7071067812.
your formula is y = 40 * (.7071067812) ^ x
note that sqrt(.5) = .7071067812.
you can graph this formula.
it looks like this:
your unrounded formula is:
y = 40 * .7071067812 ^ x
your rounded formula is:
y = 40 * .71 ^ x
your rate of decay is 50% every 2 days.
every day, the amount of sucralose in the body is .7071067812 times the amount in the previous day.
that's a loss of (1 - .7071067812) = .2928932188 * the amount of sucralose from the previous day.
on day 0 you have 40.
on day 1 you have .7071067812 * 40 = 28.28427125.
that's a loss of 40 - 28.28427125 = 11.71572875 sucralos.
that is 11.71572875 / 40 = a loss of .2928932188 = 29.28932188%.
the rounded rate of decay formula is:
y = 40 * .71 ^ x
every day, you have .71 * the amount of sucralose in your body that you had the previous day.
let me know if you have any questions.
theo