A radioactive substance decays to the formula N=8e^-0.6t , where N is the number of milligrams present after t hours. Determine the initial amount of the substance to the nearest 10th of an hour, determine the half life of the substance, and determine the number of hours for 0.3 mg to remain.
"Determine the initial amount of the substance to the nearest 10th of an hour"
HUH, since when is an amount of something measured in units of time?? (I will just ignore that part of the question)
N = 8 e^(-6t)
initial --- t=0
N = 8 e^0 = 8 mg <----- the initial amount
for half-life
.5 = e^(-.6t)
take ln of both sides
ln .5 = -.6t ln e, but lne = 1
-.6t = ln .5
t = 1.155..
the half life is 1.155 hrs or 1 hour and appr 9 minutes
.3 = 8 e^(-.6t)
.0375 = e^(-.6t)
ln both sides
ln .0375 = -.6t
t = 5.47 hrs or 5 hours and appr 28 minutes
Omg you are amazing. You don't even know. I have been sitting here trying to figure this out for hours. Thank you so much! (Idk about the time thing, that's how it's written) THANK YOU!
To determine the initial amount of the substance, we need to find the value of N when t is equal to 0.
Given the formula for decay, N = 8e^(-0.6t), we substitute t = 0:
N = 8e^(-0.6 * 0)
N = 8e^0
N = 8 * 1
N = 8
Therefore, the initial amount of the substance is 8 milligrams.
To determine the half-life of the substance, we need to find the time it takes for the amount to decrease by half, or N to be equal to half the initial amount (4 mg).
Given N = 8e^(-0.6t), we substitute N = 4 and solve for t:
4 = 8e^(-0.6t)
Dividing both sides by 8:
0.5 = e^(-0.6t)
Taking the natural logarithm (ln) of both sides:
ln(0.5) = -0.6t
Solving for t:
t = ln(0.5) / -0.6
Using a calculator, we find:
t ≈ 1.1513 hours
Therefore, the half-life of the substance is approximately 1.2 hours (to the nearest tenth).
To determine the number of hours for the substance to decay to 0.3 mg, we need to solve the equation N = 0.3 for t:
0.3 = 8e^(-0.6t)
Dividing both sides by 8:
0.0375 = e^(-0.6t)
Taking the natural logarithm (ln) of both sides:
ln(0.0375) = -0.6t
Solving for t:
t = ln(0.0375) / -0.6
Using a calculator, we find:
t ≈ 3.131 hours
Therefore, it takes approximately 3.1 hours (to the nearest tenth) for the substance to decay to 0.3 mg.