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: Substitute values into the formula

Given \(m = 50\), \(t = 10\), \(h=5.3\). The formula is \(f(t)=m(0.5)^{\frac{t}{h}}\). Substituting, we get \(f(10)=50(0.5)^{\frac{10}{5.3}}\).

Step2: Calculate the value

Using a calculator, \(f(10)=50\times(0.5)^{\frac{10}{5.3}}\approx50\times0.27\approx13.5\)

Answer:

\(f(10) = 50(0.5)^{\frac{10}{5.3}}; 13.5\space mg\) (first option)