Element X decays radioactively with a half life of 6 minutes. If there are 680 grams of Element X, how long, *to the nearest tenth of a minute*, would it take the element to decay to 276 grams?

y, equals, a, left parenthesis, point, 5, right parenthesis, start superscript, start fraction, t, divided by, h, end fraction, end superscript

*y*=*a*(.5)*ht*​

The formula to model radioactive decay is given by:

\[
y = a \left( 0.5 \right)^{\frac{t}{h}}
\]

Where:
- \( y \) is the final amount of the substance (276 grams in this case),
- \( a \) is the initial amount of the substance (680 grams),
- \( t \) is the time we want to find,
- \( h \) is the half-life of the substance (6 minutes for Element X).

We need to solve for \( t \), the time it takes for the element to decay from 680 grams to 276 grams.

### Step 1: Set up the equation
\[
276 = 680 \left( 0.5 \right)^{\frac{t}{6}}
\]

### Step 2: Divide both sides by 680
\[
\frac{276}{680} = \left( 0.5 \right)^{\frac{t}{6}}
\]

\[
0.40588 = \left( 0.5 \right)^{\frac{t}{6}}
\]

### Step 3: Take the natural logarithm of both sides
\[
\ln(0.40588) = \ln \left( \left( 0.5 \right)^{\frac{t}{6}} \right)
\]

\[
\ln(0.40588) = \frac{t}{6} \ln(0.5)
\]

### Step 4: Solve for \( t \)
\[
t = 6 \cdot \frac{\ln(0.40588)}{\ln(0.5)}
\]

Now, let's calculate \( t \).The time it would take for Element X to decay from 680 grams to 276 grams is approximately **7.8 minutes**.

Would you like more details on this calculation, or do you have any further questions?

Here are 5 related questions to deepen your understanding:
1. How would the result change if the half-life was different, say 10 minutes?
2. Can you derive the formula for decay starting from exponential growth equations?
3. What would be the mass of Element X after 12 minutes?
4. How can you calculate the remaining amount after any given time?
5. How does the decay rate differ for different types of radioactive elements?

**Tip:** The natural logarithm (\( \ln \)) is a crucial tool when solving exponential decay problems. Make sure you are comfortable using it in different contexts.

Element X decays radioactively with a half-life of 6 minutes. If there are 680 grams of Element X, how long, to the nearest tenth of a minute, would it take the element to decay to 276 grams?

Discover how to calculate the time for Element X to decay from 680 grams to 276 grams with a 6-minute half-life using exponential decay and logarithmic functions. A detailed step-by-step solution explains the math behind radioactive decay, suitable for Grades 9-12.

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