Please help me solve this calculus question

Question

A radioactive substance decays so that after t years, the amount remaining, expressed as A = 100e^-0.3t

a. What is the half-life for this substance?

b. What is the rate of decay when half the substance has decayed?

**please show method also

Answer 1

The basic formula for exponential decay is A=A₀e^-kt, where A₀ is the inital amount and A is the final amount. Half-life is the amount of time it takes for half of a substance to no longer be radioactive. The equation in the problem states that the starting amount is 100 (grams or some unit of mass). So if you set the final amount to half the starting amount, then solve for t, you'll have the half-life of the substance.

I'm not totally sure about part B, but since it says "rate," that makes me thinks they want the amount per unit of time. For example, if it takes 2 years for 100g of the substance to decay by half (i.e. 50g are still radioactive) then the rate of decay would be 50g per 2 years, or 25g per year. But they don't give you any units for the mass of the substance, so I can't be sure.

I hope this helps, let me know if you have other questions.