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)=5.3(0.5)^{0.2};4.6 mathrm{mg} )
( f(10)=50(0.50)^{0.53};34.6 mathrm{mg} )
( f(10)=50(0.5)^{\frac{10}{5.3}};13.5 mathrm{mg} )
( f(10)=5.3(0.5)^{5};0.2 mathrm{mg} )

Explanation:

Step1: Identificar los valores de \(m\), \(t\) y \(h\)

Dado que \(m = 50\) (masa inicial), \(t = 10\) (años) y \(h=5.3\) (semi - vida).

Step2: Sustituir en la fórmula \(f(t)=m(0.5)^{\frac{t}{h}}\)

Sustituyendo los valores, obtenemos \(f(10)=50(0.5)^{\frac{10}{5.3}}\).

Step3: Calcular \(f(10)\)

\(\frac{10}{5.3}\approx1.887\). Luego \(0.5^{1.887}=\frac{1}{2^{1.887}}\approx0.27\). Y \(50\times0.27 = 13.5\)

Answer:

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