Water is flowing from a horizontal pipe 48 feet above the...

Water is flowing from a horizontal pipe 48 feet above the ground. The falling stream of water has the shape of a parabola whose vertex (0,48) is at the end of the pipe (see figure). The stream of water strikes the ground at the point $(10 \sqrt{3}, 0)$. Find the equation of the path taken by the water.

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Key Concepts

Quadratic Function and Vertex Form

A quadratic function is typically expressed in the form y = a(x - h)² + k, where (h, k) is the vertex of the parabola. This vertex form makes it straightforward to identify the highest or lowest point of the parabola, which is essential when the vertex is a key element of the problem.

Determining Parameters Using Given Points

When additional points on the curve are known, they can be substituted into the vertex form of the quadratic function to solve for the unknown coefficient 'a'. This process of parameter determination allows one to completely define the equation of the parabola based on given conditions.

Projectile Motion and Parabolic Paths

Many physical scenarios, such as objects or fluids under the influence of gravity, follow a parabolic trajectory. Understanding that such paths can be modeled with quadratic functions is crucial, as it connects the physical phenomenon of projectile motion with its algebraic representation.

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