Question

certain radioactive material decays in such a way that the mass remaining after t years is given by the function
m(t) = 370(0.98)^t
where m(t) is measured in grams.
(a) find the mass at time t = 0.
your answer is
(b) how much of the mass remains after 15 years?
your answer is
give your answer to two decimal places.

Explanation:

Part (a)

Step1: Substitute \( t = 0 \) into \( m(t) \)

We know that any non - zero number to the power of 0 is 1. So when we substitute \( t = 0 \) into the function \( m(t)=370(0.98)^{t} \), we get \( m(0)=370\times(0.98)^{0} \).
Since \( (0.98)^{0}=1 \), then \( m(0)=370\times1 \).

Step2: Calculate the result

\( m(0) = 370 \)

Part (b)

Step1: Substitute \( t = 15 \) into \( m(t) \)

Substitute \( t = 15 \) into the function \( m(t)=370(0.98)^{t} \), we get \( m(15)=370\times(0.98)^{15} \).

Step2: Calculate \( (0.98)^{15} \)

Using a calculator, \( (0.98)^{15}\approx0.7397 \) (you can calculate this by repeatedly multiplying 0.98 fifteen times or using the power function on a calculator).

Step3: Calculate \( m(15) \)

Multiply 370 by 0.7397: \( m(15)=370\times0.7397 = 273.689\approx273.69 \) (rounded to two decimal places)

Answer:

(a) \( \boldsymbol{370} \) grams
(b) \( \boldsymbol{273.69} \) grams