Find y for a 20.5% increase in damage per bullet when compared to 100 bullets for y.

We will use the given variables: y= bullets fired x = 145 z = 100 Total damage = [(y-18)*122.5]+2025 Total damage/y = Damage per bullet

Let's solve for \( y \) given the new conditions:

1. **Calculate the Damage per Bullet for 100 Bullets**

   Given:
   \[
   y = 100
   \]
   \[
   \text{Total damage} = [(100 - 18) \times 122.5] + 2025
   \]

   Calculate the total damage for 100 bullets:
   \[
   \text{Total damage}_{100} = (82 \times 122.5) + 2025
   \]
   \[
   \text{Total damage}_{100} = 10045 + 2025
   \]
   \[
   \text{Total damage}_{100} = 12070
   \]

   Now, calculate the damage per bullet for 100 bullets:
   \[
   \text{Damage per bullet}_{100} = \frac{12070}{100} = 120.7
   \]

2. **Calculate the Damage per Bullet for \( y \) Bullets**

   Given that the damage per bullet increases by 20.5% when using \( y \) bullets, we express this increase as:
   \[
   \text{Damage per bullet}_{y} = 120.7 \times (1 + 0.205) = 120.7 \times 1.205 = 145.4435
   \]

3. **Set up the Equation for Total Damage for \( y \) Bullets**

   We need to find \( y \) such that the damage per bullet equals 145.4435:
   \[
   \text{Total damage}_{y} = [(y - 18) \times 122.5] + 2025
   \]

   The damage per bullet for \( y \) bullets is:
   \[
   \frac{\text{Total damage}_{y}}{y} = 145.4435
   \]

4. **Solve for \( y \)**

   Set up the equation:
   \[
   \frac{[(y - 18) \times 122.5] + 2025}{y} = 145.4435
   \]

   Multiply both sides by \( y \):
   \[
   [(y - 18) \times 122.5] + 2025 = 145.4435y
   \]

   Expand and simplify:
   \[
   122.5y - 2205 + 2025 = 145.4435y
   \]

   Combine like terms:
   \[
   122.5y - 180 = 145.4435y
   \]

   Isolate \( y \):
   \[
   -180 = 145.4435y - 122.5y
   \]
   \[
   -180 = 22.9435y
   \]

   Solve for \( y \):
   \[
   y = \frac{-180}{22.9435}
   \]
   \[
   y \approx -7.84
   \]

Again, \( y \) turns out to be a negative number, which is not physically meaningful in this context. It appears there might be a misinterpretation of the problem constraints or the damage calculation model. Let's reconsider the assumptions:

- The percentage increase in damage per bullet was used directly. Ensure this matches the given problem context.
- Verify if all given values and percentages are accurate.

Could there be a typo or misinterpretation in the problem statement?

### Questions for Further Clarification

1. Are the provided damage values and percentages correctly interpreted?
2. Is the 20.5% increase correctly applied to the original damage per bullet?
3. Could there be additional factors affecting the damage calculation?
4. Is there a real-world scenario or example that can verify these calculations?
5. Are there alternative methods to solve for \( y \) that we haven't considered?
6. How can we confirm the calculations step by step?
7. Could additional constraints or conditions clarify the negative \( y \)?
8. Is there a graphical method to verify the solution?

### Tip

Always cross-check the calculations and consider alternative approaches if the initial results seem unrealistic or incorrect.

Learn how to calculate a 20.5% increase in damage per bullet using mathematical formulas and equations. Explore step-by-step solutions and verify calculations for accuracy.

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