Truck Height Limit for Ellipse Overpass: 40ft Wide, 15ft High

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SUMMARY

The discussion centers on determining the maximum height of a truck that can pass under a semi-elliptical overpass, which is 15 feet high and 40 feet wide. The mathematical representation of the upper branch of the ellipse is given by the equation $\dfrac{x^2}{20^2}+\dfrac{y^2}{15^2}=1$. To find the height at a specific width of 12 feet, the formula $y=15\sqrt{1-\dfrac{x^2}{20^2}}$ is utilized, allowing for the calculation of the maximum truck height at the specified distance from the center of the overpass.

PREREQUISITES
  • Understanding of semi-elliptical geometry
  • Basic algebra and manipulation of equations
  • Knowledge of the Pythagorean theorem
  • Ability to evaluate square roots and apply them in real-world contexts
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  • Learn how to derive the equation of an ellipse from its geometric properties
  • Explore applications of ellipses in engineering and architecture
  • Study optimization problems involving geometric constraints
  • Investigate the use of calculus in determining maximum and minimum values in real-world scenarios
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Mathematicians, engineers, and anyone involved in transportation logistics or infrastructure planning who needs to assess vehicle clearance under overpasses.

Felidaei
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The question asks, "A one-way road has an overpass in the form of a semi-ellipse, 15 ft high at the center, and 40 ft wide. Assuming a truck is 12 ft wide, what is the tallest truck that can pass under the overpass?"

I don't think this is a super complicated question yet it proves to be too confusing for my brain >_<
Any kind of help would be appreciated, thank you!
 
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Upper branch of the ellipse $\dfrac{x^2}{20^2}+\dfrac{y^2}{15^2}=1$ is ...

$y=15\sqrt{1-\dfrac{x^2}{20^2}}$

Now determine $y(6)$ ...
 

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