https://file.aiquickdraw.com/mathbot/file/1725457375828npcuezpk.jpg

To solve this problem, we need to find the maximum shear stress in the given element based on the provided stresses. The image shows a rectangular element subjected to different normal and shear stresses.

Given:
- \( \sigma_x = 10 \, \text{MPa} \) (Normal stress in the x-direction)
- \( \sigma_y = -15 \, \text{MPa} \) (Normal stress in the y-direction)
- \( \tau_{xy} = 12 \, \text{MPa} \) (Shear stress)

### Maximum Shear Stress Calculation

The formula to calculate the maximum shear stress \( \tau_{\text{max}} \) in a plane stress situation is:

\[
\tau_{\text{max}} = \sqrt{\left( \frac{\sigma_x - \sigma_y}{2} \right)^2 + \tau_{xy}^2}
\]

Now let's plug in the values:

\[
\tau_{\text{max}} = \sqrt{\left( \frac{10 \, \text{MPa} - (-15 \, \text{MPa})}{2} \right)^2 + (12 \, \text{MPa})^2}
\]

\[
\tau_{\text{max}} = \sqrt{\left( \frac{10 + 15}{2} \right)^2 + 12^2}
\]

\[
\tau_{\text{max}} = \sqrt{\left( \frac{25}{2} \right)^2 + 12^2}
\]

\[
\tau_{\text{max}} = \sqrt{(12.5)^2 + 12^2}
\]

\[
\tau_{\text{max}} = \sqrt{156.25 + 144}
\]

\[
\tau_{\text{max}} = \sqrt{300.25}
\]

\[
\tau_{\text{max}} \approx 17.33 \, \text{MPa}
\]

### Conclusion
The maximum shear stress in the element is approximately \( 17.33 \, \text{MPa} \).

So, the correct option would be **16.1 MPa** (closest to the calculated value).

Would you like further details or explanations on any part of this solution?

### Relative Questions:
1. How is the maximum shear stress formula derived from principal stresses?
2. What is the significance of positive and negative stress values in these calculations?
3. How would you determine the principal stresses in this case?
4. What other methods can be used to find the maximum shear stress?
5. How does the presence of shear stress influence the failure of materials?

### Tip:
Remember, the direction of shear stress is as important as its magnitude, especially when analyzing the potential failure modes of materials.

Find the maximum shear stress in the given element based on the provided normal and shear stresses.

Learn how to calculate the maximum shear stress in a rectangular element subjected to normal and shear stresses. This problem involves the application of the plane stress formula and Mohr's Circle, ideal for undergraduate engineering students.

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