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50POINTS- Cesium-137 has a half-life of 30 years. Find the amount of cesium-137 left from a 100-milligram sample after 180 years, round to the nearest hundredth. Include units.

Respuesta :

To find the amount of cesium-137 left after 180 years, we can use the half-life formula.

Since the half-life of cesium-137 is 30 years, we can calculate the number of half-lives that have passed in 180 years by dividing 180 by 30.

180 years / 30 years = 6 half-lives

Each half-life reduces the amount of cesium-137 by half. So, after 6 half-lives, the amount remaining will be:

100 milligrams / (2^6) = 100 milligrams / 64 = 1.5625 milligrams

Rounding to the nearest hundredth, the amount of cesium-137 left from a 100-milligram sample after 180 years is approximately 1.56 milligrams.
msm555

Answer:

1.56 milligram

Step-by-step explanation:

To find the amount of cesium-137 left after 180 years, we can use the formula for exponential decay:

[tex]\Large\boxed{\boxed{\sf N(t) = N_0 \times \left(\dfrac{1}{2}\right)^{\frac{t}{T}}}} [/tex]

Where:

  • [tex]\sf N(t) [/tex] is the amount of substance at time [tex]\sf t [/tex]
  • [tex]\sf N_0 [/tex] is the initial amount of substance
  • [tex]\sf T [/tex] is the half-life of the substance
  • [tex]\sf t [/tex] is the time that has passed

Given:

  • [tex]\sf N_0 = 100 [/tex] milligrams (initial amount)
  • [tex]\sf T = 30 [/tex] years (half-life)
  • [tex]\sf t = 180 [/tex] years

Substituting these values into the formula:

[tex]\sf N(180) = 100 \times \left(\dfrac{1}{2}\right)^{\frac{180}{30}} [/tex]

[tex]\sf N(180) = 100 \times \left(\dfrac{1}{2}\right)^6 [/tex]

[tex]\sf N(180) = 100 \times \left(\dfrac{1}{64}\right) [/tex]

[tex]\sf N(180) = \dfrac{100}{64} [/tex]

[tex]\sf N(180) \approx 1.5625 [/tex]

[tex]\sf N(180) \approx 1.56\textsf{(in nearest hundredth)}[/tex]

So, after 180 years, there are approximately 1.56 milligrams of cesium-137 left from the initial 100-milligram sample.

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