The exponential formula for the half-life of a radioactive isotope is
y=y0ekt,
where y is the amount of the isotope remaining after t years, y0 is the initial amount of the isotope, k is the decay constant, and e is the transcendental number approximately equal to 2.71828.
How can you rearrange the given formula to correctly find y0?
divide by e^(kt) ... y / [e^(kt)] = y0
let t = 0
e^0 = 1
so
y0 = y at t = 0
or if you know y and t
then
y0= y / e^kt = y e^-kt
HOWEVER , That is NOT the equation for the half life.
k by the way is a negative number as you wrote the equation
y = y0 e^kt
for half life
1/2 = y/y0 = e^kt
ln 0.5 = k t
-0.693 = k t
t = -.693 /k when it is half gone
To rearrange the formula to find y0, we need to isolate y0 on one side of the equation. Here's how you can do it:
1. Start with the original formula: y = y0e^(kt).
2. Divide both sides of the equation by e^(kt): y / e^(kt) = y0.
3. Simplify the right side of the equation: y0 = y / e^(kt).
Now, you have the rearranged formula to find y0. Simply substitute the known values for y (the amount of the isotope remaining) and t (the number of years) into the equation to calculate y0.
Remember that the value of e is approximately 2.71828, and k is the decay constant specific to the radioactive isotope you are working with.