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At the beginning of an experiment, a scientist has 248 grams of radioactive goo. After 120 minutes, her sample has decayed to 15.5 grams.

What is the half-life of the goo in minutes?

Find a formula for G(t), the amount if goo remaining at time t. G(t) =

Respuesta :

msm555

Answer:

30 minutes

Step-by-step explanation:

The half-life ([tex] t_{\frac{1}{2}} [/tex]) of a substance is the time it takes for half of the substance to decay. The half-life can be found using the formula:

[tex] N(t) = N_0 \left( \dfrac{1}{2} \right)^{\dfrac{t}{t_{\frac{1}{2}}}} [/tex]

where:

  • [tex] N(t) [/tex] is the amount of the substance remaining at time [tex] t [/tex],
  • [tex] N_0 [/tex] is the initial amount of the substance,
  • [tex] t [/tex] is the time elapsed,
  • [tex] t_{\dfrac{1}{2}} [/tex] is the half-life of the substance.

In this case, the initial amount of radioactive goo ([tex] N_0 [/tex]) is 248 grams, and after 120 minutes, the remaining amount ([tex] N(t) [/tex]) is 15.5 grams.

[tex] 15.5 = 248 \left( \dfrac{1}{2} \right)^{\dfrac{120}{t_{\frac{1}{2}}}} [/tex]

Now, solve for [tex] t_{\frac{1}{2}} [/tex]:

[tex] \dfrac{15.5}{248} = \left( \dfrac{1}{2} \right)^{\dfrac{120}{t_{\frac{1}{2}}}} [/tex]

[tex] 0.0625 = \left( \dfrac{1}{2} \right)^{\dfrac{120}{t_{\frac{1}{2}}}} [/tex]

[tex] \left(\dfrac{1}{2}\right)^4 = \left( \dfrac{1}{2} \right)^{\dfrac{120}{t_{\frac{1}{2}}}} [/tex]

Since we have the same base, we can compare the power

So,

[tex] 4 = \dfrac{120}{t_{\frac{1}{2}}} [/tex]

Solve for [tex] t_{\frac{1}{2}} [/tex]:

[tex] t_{\frac{1}{2}} =\dfrac{ 120}{4}[/tex]

[tex] t_{\frac{1}{2}} = 30[/tex]

The half-life of the goo is 30 minutes.

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