Question

A sample of radioactive material disintegrates from 6 to 4 grams in 100 days. After how many days will just 3 grams remain?

Answer 1

Answer:

150 days

Step-by-step explanation:

6-4=2

100/2=50

50*3=150

Answer 2

The number of days for the radioactive material to disintegrate to 3 grams is 173.077 days.

The rate of disintegration varies directly proportional to the quantity of the material.

As such, we can say:

$\mathbf{=\dfrac{dN}{dt}\ \alpha \ N}$

$\mathbf{\implies \dfrac{dN}{N}\ = k dt}$

Taking the integral form;

$\mathbf{\implies \int \dfrac{dN}{N}\ =\int k dt}$

$\mathbf{\implies In N =kt+ C---- (1)}$

When t = 0, N = 6 grams

In(6) = C

∴

When t = 100, N = 4 grams

In (4) = 100k + In6

100 k = 1n (4) - In(6)

$\mathbf{100 k = In (\dfrac{4}{6})}$

$\mathbf{k = \dfrac{1}{100} In(\dfrac{4}{6})}$

∴

From equation (1):

$\mathbf{In N = \dfrac{t}{100} In(\dfrac{4}{6})+ In 6}$

when,

• n = 3 grams; we have:

$\mathbf{In (3) = \dfrac{t}{100} In(\dfrac{4}{6})+ In 6}$

$\mathbf{\implies \dfrac{t}{100} In(\dfrac{4}{6}) = In \dfrac{ 3}{ 6}}$

$\mathbf{t = 100\times \Big ( \dfrac{In (\dfrac{ 3}{ 6})}{ In(\dfrac{4}{6}) }\Big) }$

$\mathbf{t = 100\times \Big ( \dfrac{0.69314}{ 0.40048}\Big) }$

t = 173.077 days

Therefore, the number of days for the radioactive material to disintegrate to 3 grams is 173.077 days.

Learn more about radioactive materials here:

https://brainly.com/question/24339152?referrer=searchResults