Sure! Let's solve this step-by-step.
### Problem Analysis
We are given:
- An initial amount of radioactive material: 180 grams
- A half-life of 1 hour, meaning the material decreases by half every hour
- We need to determine the amount of radioactive material left after 3 hours.
### Step-by-Step Solution
1. Identify the parameters for the exponential decay formula.
The general form of the exponential decay formula is:
[tex]\[ y = a \left( \frac{1}{2} \right)^{\frac{x}{b}} \][/tex]
Where:
- [tex]\( y \)[/tex] is the remaining amount of material after time [tex]\( x \)[/tex]
- [tex]\( a \)[/tex] is the initial amount of material
- [tex]\( \frac{1}{2} \)[/tex] represents the half-life decay factor
- [tex]\( x \)[/tex] is the elapsed time
- [tex]\( b \)[/tex] is the half-life period
2. Substitute the given values into the formula.
- Initial amount [tex]\( a = 180 \)[/tex] grams
- Half-life period [tex]\( b = 1 \)[/tex] hour
- Elapsed time [tex]\( x = 3 \)[/tex] hours
The formula becomes:
[tex]\[ y = 180 \left( \frac{1}{2} \right)^{\frac{3}{1}} \][/tex]
3. Simplify the exponent.
[tex]\[ y = 180 \left( \frac{1}{2} \right)^3 \][/tex]
4. Calculate the decay factor raised to the power.
[tex]\[ \left( \frac{1}{2} \right)^3 = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \][/tex]
5. Now substitute this back into the equation.
[tex]\[ y = 180 \times \frac{1}{8} \][/tex]
6. Perform the multiplication.
[tex]\[ y = 180 \times 0.125 = 22.5 \][/tex]
### Answer
After 3 hours, there will be [tex]\( 22.5 \)[/tex] grams of the radioactive material left.
To summarize:
- The value for [tex]\( x \)[/tex], the elapsed time, is [tex]\( 3 \)[/tex] hours, filling in the first blank.
- The equation we derived is [tex]\( y = 180 \left( \frac{1}{2} \right)^3 \)[/tex].
- Substituting the numeric values yields [tex]\( y = 22.5 \)[/tex].
So, you should fill in:
1. [tex]\( x = 3 \)[/tex]
2. [tex]\( y = 22.5 \)[/tex]
