Final answer:
The expression for the amount of a certain radioactive substance as a function of time with a half-life of 14 days and an initial amount of 2.5 grams is A(t) = 2.5 * (1/2)^(t/14). To find when there will be less than 1 gram remaining, we solve the equation 1 = 2.5 * (1/2)^(t/14) and approximate t to be approximately 16.22 days.
Step-by-step explanation:
The expression for the amount of a certain radioactive substance as a function of time is given by option A) A(t) = 2.5 * (1/2)^(t/14). This equation represents exponential decay, where the initial amount is multiplied by the decay factor (1/2) raised to the power of the elapsed time divided by the half-life.
To determine when there will be less than 1 gram remaining, we can set the expression equal to 1 and solve for t:
1 = 2.5 * (1/2)^(t/14)
By taking the logarithm of both sides, we can isolate the exponent t/14:
t/14 = log(base 1/2)(1/2.5)
Using the logarithmic property that log(base b)(b^x) = x, we get:
t/14 = log(base 1/2)(1/2.5) = -log(base 2)(2.5)
Multiplying both sides by 14 and dividing by log(base 2)(2.5), we find:
t = -14 * log(base 2)(2.5)
Using a calculator, we can approximate the value of t to be approximately 16.22 days.