To solve for the operating $x$-value where $y = 45'0''$, given that the rectangle is inscribed in a right triangle with a base of $35'3''$, height of $118'0''$, and hypotenuse $120'6''$, we need to use the properties of similar triangles.

Step-by-Step Solution:

1. Convert all dimensions to inches:

   • $\text{Base} = 35'3'' = 35 \times 12 + 3 = 423 \, \text{inches}$,
   • $\text{Height} = 118'0'' = 118 \times 12 = 1416 \, \text{inches}$,
   • $\text{Hypotenuse} = 120'6'' = 120 \times 12 + 6 = 1446 \, \text{inches}$,
   • $y = 45'0'' = 45 \times 12 = 540 \, \text{inches}$.

2. Identify relationships: The rectangle is inscribed, so the top corner of the rectangle touches the hypotenuse of the triangle. The triangles formed are similar, meaning the sides are proportional.

   The relation between the $x$-coordinate (base of the smaller triangle) and $y$-coordinate (height of the smaller triangle) can be derived using the slope of the hypotenuse: $\text{Slope of hypotenuse} = \frac{\text{Height}}{\text{Base}} = \frac{1416}{423}.$

3. Proportionality: At any point, $x$ and $y$ satisfy the equation of the hypotenuse, which can be written as: $y = \text{Slope} \times (423 - x).$ Substituting the slope: $y = \frac{1416}{423} \cdot (423 - x).$

4. Substitute $y = 540 \, \text{inches}$: $540 = \frac{1416}{423} \cdot (423 - x).$ Simplify: $540 = \frac{1416}{423} \cdot 423 - \frac{1416}{423} \cdot x.$ $540 = 1416 - \frac{1416}{423} \cdot x.$ $\frac{1416}{423} \cdot x = 1416 - 540.$ $\frac{1416}{423} \cdot x = 876.$

5. Solve for $x$: $x = \frac{876 \cdot 423}{1416}.$ Simplify: $x = \frac{370548}{1416}.$ $x = 261.6 \, \text{inches}.$

6. Convert $x$ back to feet and inches: $x = 261.6 \, \text{inches}.$ Convert: $\text{Feet} = \lfloor 261.6 / 12 \rfloor = 21' \quad \text{(Whole feet)}.$ $\text{Inches} = 261.6 - (21 \times 12) = 261.6 - 252 = 9.6''.$

   So, $x \approx 21'9.6''$.

Final Answer:

The operating $x$-value is approximately 21'9.6''.

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Would you like further details or explanations?

Related Questions:

1. How can the slope of the hypotenuse be applied to other geometric problems?
2. What if the rectangle had a different height, such as $y = 60'0''$?
3. How do similar triangles help in solving real-world engineering problems?
4. What role does the hypotenuse play in calculating proportionality?
5. Can we derive the formula for $x$ in terms of any given $y$-value?

Tip:

Always convert all measurements to a single unit before solving for simplicity and accuracy.