A semi elliptic archway has a height of 20 feet and a width...

Name: Problem 44, Chapter 10: Introductory and Intermediate Algebra for College Students
Uploaded: 2020-08-01T06:26:22-08:00
Duration: 3 min 7 s
Channel: Suman Saurav Thakur
Description: A semi elliptic archway has a height of 20 feet and a width of 50 feet, as shown in the figure. Can a truck 14 feet high and 10 feet wide drive under the archway without going into the other lane?

A semi elliptic archway has a height of 20 feet and a width of 50 feet, as shown in the figure. Can a truck 14 feet high and 10 feet wide drive under the archway without going into the other lane?

Key Concepts

Ellipse

An ellipse is a conic section defined as the set of all points such that the sum of their distances from two fixed points (the foci) remains constant. In analytic geometry, understanding the properties of an ellipse—including its major and minor axes and its standard equation—is key to analyzing curves and arches.

Standard Equation of an Ellipse

The standard form of the ellipse equation, typically written as (x²/a²) + (y²/b²) = 1, provides a mathematical description of the ellipse. This equation allows for determining the coordinates of any point on the ellipse, which is essential for problems that require calculating height or clearance at a given horizontal position.

Substitution in Equations

Substitution is a common algebraic technique where a specific value is inserted into an equation to solve for another variable. In the context of an ellipse, substituting a known horizontal distance into the standard equation can yield the corresponding vertical coordinate, helping assess if an object fits under a curved structure.

Clearance Analysis

Clearance analysis uses geometry to determine whether an object can pass through or fit underneath a particular structure by comparing the object’s dimensions with the available space. By calculating the height of the arch at specific horizontal points, one can verify that an object like a vehicle meets the necessary clearance requirements.

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