To find the total electric flux through the rectangular box, you need to sum the electric fluxes through the six sides of the box. Here is the step-by-step process:
1. Identify the given electric fluxes:
[tex]\[
\Phi_1 = 143.6 \, \text{N} \cdot \text{m}^2 / \text{C}
\][/tex]
[tex]\[
\Phi_2 = 266.0 \, \text{N} \cdot \text{m}^2 / \text{C}
\][/tex]
[tex]\[
\Phi_3 = -342.5 \, \text{N} \cdot \text{m}^2 / \text{C}
\][/tex]
[tex]\[
\Phi_4 = 191.1 \, \text{N} \cdot \text{m}^2 / \text{C}
\][/tex]
[tex]\[
\Phi_5 = -119.5 \, \text{N} \cdot \text{m}^2 / \text{C}
\][/tex]
[tex]\[
\Phi_6 = 453.4 \, \text{N} \cdot \text{m}^2 / \text{C}
\][/tex]
2. Sum all the electric fluxes:
[tex]\[
\text{Total Flux} = \Phi_1 + \Phi_2 + \Phi_3 + \Phi_4 + \Phi_5 + \Phi_6
\][/tex]
3. Write down the addition of each flux term:
[tex]\[
\text{Total Flux} = 143.6 + 266.0 + (-342.5) + 191.1 + (-119.5) + 453.4
\][/tex]
4. Combine the positive flux values:
[tex]\[
143.6 + 266.0 + 191.1 + 453.4 = 1054.1
\][/tex]
5. Combine the negative flux values:
[tex]\[
-342.5 + (-119.5) = -462.0
\][/tex]
6. Add the combined positive and negative results:
[tex]\[
1054.1 + (-462.0) = 592.1
\][/tex]
Therefore, the total electric flux through the rectangular box is:
[tex]\[
\boxed{592.1 \, \text{N} \cdot \text{m}^2 / \text{C}}
\][/tex]
