Radium decays according to the function

where represents the quantity remaining after years and is the decay
constant 0.00043 . What is the half-life of radium?
Calculate how long it will take for 120 g of radium to decay to 60 g . Round your
answer to the nearest year.
A. 12 years
B. 806 years
C. 1612 years
D. 55,813 years

Ask by Dickson Cole. in the United States

Dec 15,2024

Question

Radium decays according to the function

where represents the quantity remaining after years and is the decay
constant 0.00043 . What is the half-life of radium?
Calculate how long it will take for 120 g of radium to decay to 60 g . Round your
answer to the nearest year.
A. 12 years
B. 806 years
C. 1612 years
D. 55,813 years

Ask by Dickson Cole. in the United States

Dec 15,2024

UpStudy AI · Accepted Answer

To find the half-life of radium, we start with the decay function:

$$
Q(t) = Q_0 e^{-kt}
$$

The half-life $$ t_{1/2} $$ is the time it takes for the quantity to reduce to half of its initial value. Therefore, we set $$ Q(t_{1/2}) = \frac{Q_0}{2} $$:

$$
\frac{Q_0}{2} = Q_0 e^{-kt_{1/2}}
$$

Dividing both sides by $$ Q_0 $$ (assuming $$ Q_0 \neq 0 $$) gives:

$$
\frac{1}{2} = e^{-kt_{1/2}}
$$

Taking the natural logarithm of both sides:

$$
\ln\left(\frac{1}{2}\right) = -kt_{1/2}
$$

We know that $$ \ln\left(\frac{1}{2}\right) = -\ln(2) $$, so we can rewrite the equation as:

$$
-\ln(2) = -kt_{1/2}
$$

This simplifies to:

$$
t_{1/2} = \frac{\ln(2)}{k}
$$

Substituting $$ k = 0.00043 $$:

$$
t_{1/2} = \frac{\ln(2)}{0.00043}
$$

Calculating $$ \ln(2) $$:

$$
\ln(2) \approx 0.693147
$$

Now substituting this value into the equation:

$$
t_{1/2} = \frac{0.693147}{0.00043} \approx 1612.1 \text{ years}
$$

Rounding to the nearest year, the half-life of radium is approximately:

$$
t_{1/2} \approx 1612 \text{ years}
$$

Next, we need to calculate how long it will take for 120 g of radium to decay to 60 g. We set up the equation:

$$
60 = 120 e^{-kt}
$$

Dividing both sides by 120 gives:

$$
\frac{1}{2} = e^{-kt}
$$

Taking the natural logarithm of both sides:

$$
\ln\left(\frac{1}{2}\right) = -kt
$$

Again, using $$ \ln\left(\frac{1}{2}\right) = -\ln(2) $$:

$$
-\ln(2) = -kt
$$

This simplifies to:

$$
t = \frac{\ln(2)}{k}
$$

Since we already calculated $$ t_{1/2} $$ as $$ 1612 $$ years, we find that it will take:

$$
t \approx 1612 \text{ years}
$$

Thus, the answer is:

C. 1612 years
The half-life of radium is approximately 1612 years. It will take about 1612 years for 120 g of radium to decay to 60 g.

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