Solution: We are given $ r(t) = \sqrt{(t - 8)^2} $.

["Understanding and Solving the Function: ( r(t) = \sqrt{(t - 8)^2} )", "When analyzing real-world problems involving distances or magnitudes, functions of the form ( r(t) = \sqrt{(t - a)^2} ) appear frequently. In this specific case, the function is ( r(t) = \sqrt{(t - 8)^2} ). This article explains the meaning, simplification, domain, range, and key solutions associated with this expression—offering a clear mathematical foundation useful for students, educators, and enthusiasts alike.", "---", "### What is ( r(t) = \sqrt{(t - 8)^2} )?", "The function ( r(t) = \sqrt{(t - 8)^2} ) represents the Euclidean distance of a point ( t ) from the fixed point ( 8 ) on the number line. Since the square root of a square returns the absolute value, this function simplifies elegantly.", "---", "### Simplifying the Function", "By a fundamental identity in algebra:", "[\n\sqrt{x^2} = |x|\n]", "Applying this property to our function:", "[\nr(t) = \sqrt{(t - 8)^2} = |t - 8|\n]", "Thus, the simplified form of ( r(t) ) is the absolute value function centered at ( t = 8 ).", "---", "### Geometric Interpretation", "Imagine the number line with 8 marked clearly. The expression ( |t - 8| ) measures how far ( t ) is from 8, regardless of direction:", "- If ( t \geq 8 ), then ( r(t) = t - 8 ) (positive distance)\n- If ( t < 8 ), then ( r(t) = 8 - t ) (distance measured backward)", "Visually, the graph is a V-shaped "corner" at ( t = 8 ), forming two linear regions meeting at zero slope.", "---", "### Domain and Range of the Function", "- Domain:\n The square root is defined for all real numbers. So,\n [\n \ ext{Domain: } (-\infty, \infty)\n ]", "- Range:\n Since the absolute value outputs non-negative numbers,\n [\n \ ext{Range: } [0, \infty)\n ]", "---", "### Key Solutions and Equations Involving ( r(t) )", "Understanding conditions where ( r(t) ) takes specific values helps solve real-life problems. Consider some common equations:", "1. Solving ( r(t) = 0 ):\n [\n |t - 8| = 0 \implies t = 8\n ]\n The point ( t = 8 ) is exactly at 8—distance zero.", "2. Solving ( r(t) = k ) (where ( k \geq 0 )):\n [\n |t - 8| = k \implies t - 8 = k \quad \ ext{or} \quad t - 8 = -k\n ]\n Thus,\n [\n t = 8 + k \quad \ ext{or} \quad t = 8 - k\n ]\n This gives two solutions symmetric about 8.", "---", "### Applications in Real Life", "- Distance Measurements: The function models distance from a fixed reference (e.g., 8 o’clock on a clock, 8 km from a landmark).\n- Physics: Used in resolving directional displacement into magnitude.\n- Engineering & Computer Science: Absolute value functions appear in error analysis, optimization, and signal processing.", "---", "### Practical Example", "Suppose you’re tracking temperature deviations from a target of 8°C:", "| Time ( t ) (°C) | Distance ( r(t) )\n|-------------------|----------------------|\n| 7 | ( |7 - 8| = 1 ) |\n| 8 | ( |8 - 8| = 0 ) |\n| 9 | ( |9 - 8| = 1 ) |\n| 6 | ( |6 - 8| = 2 ) |\n| — | Measures how far off 8°C |", "This mirrors real-world cooling or heating trends.", "---", "### Summary", "The function ( r(t) = \sqrt{(t - 8)^2} ) simplifies to ( |t - 8| ), a cornerstone absolute value expression. Its key traits include:", "- Defined for all real ( t )\n- Outputs non-negative values\n- Symmetric about ( t = 8 )\n- Useful in modeling distance, error, and change across disciplines", "Mastering this function equips learners to tackle more complex problems involving magnitude and absolute differences.", "---", "### Want to Go Deeper?", "- Explore piecewise representations of absolute value functions.\n- Investigate limits and continuity at ( t = 8 ).\n- Apply ( r(t) ) in optimization or geometry contexts.", "Understanding ( r(t) = \sqrt{(t - 8)^2} = |t - 8| ) opens doors to deeper mathematical modeling and problem-solving.", "---", "Keywords: ( r(t) = \sqrt{(t - 8)^2} ), absolute value function, distance function, simplification, domain and range, real-world applications, math education.\nMeta Description: Learn how ( r(t) = \sqrt{(t - 8)^2} ) simplifies to ( |t - 8| ), its domain and range, and real-life applications in math, science, and engineering."]