Question

the amount of a radioactive substance remaining after t years is given by the function $f(t)=m(0.5)^{\frac{t}{h}}$, where m is the initial mass and h is the half - life in years. cobalt - 60 has a half - life of about 5.3 years. which equation gives the mass of a 50 mg cobalt - 60 sample remaining after 10 years, and approximately how many milligrams remain?
$f(10)=50(0.5)^{\frac{10}{5.3}};13.5$ mg
$f(10)=50(0.50)^{0.53};34.6$ mg
$f(10)=5.3(0.5)^{5};0.2$ mg
$f(10)=5.3(0.5)^{0.2};4.6$ mg

Explanation:

Step1: Identify values for formula

Given $m = 50$ (initial mass), $t = 10$ (time in years), $h=5.3$ (half - life in years). The formula for the amount of radioactive substance remaining is $f(t)=m(0.5)^{\frac{t}{h}}$. Substituting the values, we get $f(10)=50(0.5)^{\frac{10}{5.3}}$.

Step2: Calculate the value

First, calculate the exponent $\frac{10}{5.3}\approx1.8868$. Then, $(0.5)^{1.8868}\approx0.27$. Multiply by the initial mass: $50\times0.27 = 13.5$ mg.

Answer:

A. $f(10)=50(0.5)^{\frac{10}{5.3}}; 13.5$ mg