Explanation:
your teacher could have explained this better by telling us that A(t) is the remaining amount of the original isotope.
(a)
we start with 200 g and let it do its thing for 25 years.
all we need to do is enter the number of years into the function formula (as this is a college question you are able to work with functions, right ?) and calculate :
A(25) = 200e^(-0.054 × 25) = 200e^-1.35 =
= 51.84805213... g ≈ 51.85 g
after 25 years, about 51.85 g of the sample will be left.
assuming you have a calculator (at least on your computer) that was really "difficult", don't you think ?
(b)
half of the original amount is 200/2 = 100 g.
so, we need to find the value for t so that A(t) = 100.
100 = 200e^(-0.054 × t)
100/200 = 1/2 = 0.5 = e^(-0.054 × t)
ln(0.5) = -0.054 × t
t = ln(0.5) / -0.054 = 12.8360589... years
I assume you need to round to the nearest hundredth here as well :
after about 12.84 years the sample will decay to half of the original amount.