a certain radioactive substance is decaying so that at time t, measured in years, the amount of the substance, in grams, is given by the function f(t)=3e^-3t. What is the rate of decay of the substance after half a year:
I first found the derivative of f(t)=3e^-3t which is f'(t)=-9e^-3t. Then half a year would be 365/2 which is 182.5 days. When I substitute that in for t, i would get an answer of 0. What am i doing wrong?
Why are you using days when t was defined in years
so t = 1/2
f ' (1/2) = -9 e^(-3(1/2))
= -9 e^(-1.5)
= appr -2.008
Ah i see thanks!
You made a small mistake when substituting the value for half a year. Half a year corresponds to t = 0.5 years, not 182.5 days.
To find the rate of decay at half a year, substitute t = 0.5 into the derivative function f'(t) = -9e^(-3t):
f'(0.5) = -9e^(-3*0.5)
= -9e^(-1.5)
Now, you can calculate the numerical value for the rate of decay.
You are calculating the rate of decay correctly, but there seems to be a mistake when substituting the value of t. Half a year is indeed 182.5 days, but you need to convert it to years to match the units in the given function.
To convert 182.5 days to years, you divide by the number of days in a year. Assuming there are 365 days in a year, you can use the following conversion:
182.5 days ÷ 365 days/year = 0.5 years
So, after half a year, you should substitute t = 0.5 in the derivative:
f'(0.5) = -9e^(-3(0.5))
= -9e^(-1.5)
To calculate this value, you can use a calculator or approximate the value of e^(-1.5) as 0.22313016014. Therefore:
f'(0.5) ≈ -9 * 0.22313016014
≈ -2.0081714413
Hence, the rate of decay of the substance after half a year is approximately -2.008 grams per year.