To determine the amount of energy produced by converting [tex]\(6.8 \times 10^{-6} \, \text{kg}\)[/tex] of matter into energy using the equation [tex]\(E = mc^2\)[/tex], we will follow these steps:
1. Identify the given values:
- Mass loss [tex]\( \Delta m = 6.8 \times 10^{-6} \, \text{kg} \)[/tex]
- Speed of light [tex]\( c = 3 \times 10^8 \, \text{m/s} \)[/tex]
2. Substitute these values into the equation [tex]\(E = mc^2\)[/tex]:
[tex]\[
E = (6.8 \times 10^{-6} \, \text{kg}) \times (3 \times 10^8 \, \text{m/s})^2
\][/tex]
3. Calculate the energy [tex]\(E\)[/tex]:
[tex]\[
E = 6.8 \times 10^{-6} \, \text{kg} \times (3 \times 10^8 \, \text{m/s})^2
\][/tex]
4. First, calculate [tex]\( (3 \times 10^8 \, \text{m/s})^2 \)[/tex]:
[tex]\[
(3 \times 10^8 \, \text{m/s})^2 = 9 \times 10^{16} \, \text{m}^2/\text{s}^2
\][/tex]
5. Now, multiply this result with the mass loss:
[tex]\[
E = 6.8 \times 10^{-6} \, \text{kg} \times 9 \times 10^{16} \, \text{m}^2/\text{s}^2
\][/tex]
6. Perform the multiplication:
[tex]\[
E = 6.8 \times 9 \times 10^{-6 + 16} \, \text{kg} \times \text{m}^2/\text{s}^2
\][/tex]
[tex]\[
E = 61.2 \times 10^{10} \, \text{kg} \times \text{m}^2/\text{s}^2
\][/tex]
7. Simplify the exponent:
[tex]\[
E = 6.12 \times 10^{11} \, \text{kg} \times \text{m}^2/\text{s}^2
\][/tex]
The energy produced by converting [tex]\(6.8 \times 10^{-6} \, \text{kg}\)[/tex] of matter into energy in this fusion reaction is [tex]\(6.12 \times 10^{11} \, \text{J}\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{E = 6.12 \times 10^{11} \, \text{J}} \][/tex]
