3A software engineer develops an augmented reality application where a virtual object’s position in 3D space is defined by the parametric equations $x(t) = 2t + 1$, $y(t) = -t + 4$, $z(t) = 3t - 2$. At what value of $t$ does the object’s path cross the plane $x + y + z = 15$?

["Title: How 3A Software Engineer Built an AR App Using Parametric Equations and Plane Intersection", "In the rapidly evolving world of augmented reality (AR), programming precision meets creative visualization—exactly what 3A software engineer did when developing a real-time 3D application. Central to the app’s functionality was defining the exact path of a virtual object in 3D space using parametric equations. This article explains how the engineer leveraged mathematical modeling, specifically solving for when the object’s trajectory intersects a designated plane, and uncovered the precise value of parameter $ t $ at which this occurs.", "### Understanding the 3D Path of the Virtual Object", "The engineer modeled the 3D position of the virtual object using the following parametric equations:\n- $ x(t) = 2t + 1 $\n- $ y(t) = -t + 4 $\n- $ z(t) = 3t - 2 $", "These equations describe a smooth, linear path through space as $ t $ increases. Each coordinate evolves independently with time, allowing realistic animation and interaction within augmented reality environments.", "### The Challenge: Finding the Intersection with a Plane", "For AR applications, precise environmental interaction is vital—one key moment is when the virtual object crosses a specific plane. In this case, the object must intersect the plane defined by:\n$$\nx + y + z = 15\n$$", "The engineer’s task was to determine the value of $ t $ at which the sum of the parametric coordinates satisfies this equation.", "### Solving for $ t $: The Step-by-Step Breakdown", "To find the intersection point, substitute $ x(t) $, $ y(t) $, and $ z(t) $ into the plane equation:\n$$\nx(t) + y(t) + z(t) = (2t + 1) + (-t + 4) + (3t - 2) = 15\n$$", "Combine like terms:\n- $ 2t - t + 3t = 4t $\n- $ 1 + 4 - 2 = 3 $", "So the equation becomes:\n$$\n4t + 3 = 15\n$$", "Solving for $ t $:\n$$\n4t = 12 \implies t = 3\n$$", "### Significance in AR Development", "At $ t = 3 $, the virtual object crosses the plane $ x + y + z = 15 $ in real time—an essential moment for synchronizing virtual content with physical space. This calculation helps ensure accurate timing and positioning, improving user immersion and interaction fidelity.", "### Conclusion", "The intersection occurs precisely at $ t = 3 $, demonstrating how mathematical modeling underpins real-world AR applications. By defining the object’s trajectory with parametric equations and solving for spatial-plane intersections, 3A software engineer showcased the powerful synergy between programming and applied geometry. This achievement not only advances the functionality of the AR app but also reflects the precision and innovation defining modern software development.", "For developers and AR enthusiasts, understanding parametric modeling and geometric constraints like plane intersections opens new possibilities—just like the breakthrough at $ t = 3 $ opened a more seamless and responsive augmented reality experience."]