Question: A self-healing polymer film repairs itself over time, with the repair rate modeled by $ r(t) = \sqrt{(t - 8)^2} $ millimeters per hour. For how many integer values of $ t $ is the repair rate exactly 4 mm/hour?

["Understanding the Self-Healing Polymer Film’s Repair Rate: When is $ r(t) = 4 $?", "A cutting-edge self-healing polymer film has been developed to autonomously repair damage over time—a breakthrough with enormous potential for electronics, medical devices, and aerospace coatings. The repair rate of this smart material is governed by a time-dependent function:", "$$\nr(t) = \sqrt{(t - 8)^2} \quad \ ext{(in mm/hour)}\n$$", "But a crucial question arises: For how many integer values of $ t $ is the repair rate exactly 4 mm/hour?", "Let’s analyze the function to answer this precisely.", "---", "### Step 1: Simplify the Repair Rate Function", "The expression $ \sqrt{(t - 8)^2} $ simplifies using the fundamental identity:", "$$\n\sqrt{x^2} = |x|\n$$", "So,", "$$\nr(t) = |t - 8|\n$$", "This makes intuitive sense: repair occurs at a speed proportional to the absolute distance from $ t = 8 $, with $ r(t) > 0 $ for $ t <br/>\ne 8 $, and $ r(t) = 0 $ only when $ t = 8 $.", "---", "### Step 2: Solve $ |t - 8| = 4 $", "We want to find all integer values of $ t $ such that:", "$$\n|t - 8| = 4\n$$", "The equation $ |x| = a $, where $ a > 0 $, has exactly two solutions in the real numbers:", "$$\nx = a \quad \ ext{and} \quad x = -a\n$$", "Therefore,", "$$\nt - 8 = 4 \quad \Rightarrow \quad t = 12\n$$\n$$\nt - 8 = -4 \quad \Rightarrow \quad t = 4\n$$", "So the only real solutions are $ t = 4 $ and $ t = 12 $.", "Both are integers, as required by the question.", "---", "### Step 3: Determine the Number of Integer Solutions", "We are asked for the number of integer values of $ t $ satisfying the condition.", "Since only $ t = 4 $ and $ t = 12 $ yield $ r(t) = 4 $, and both are integers, there are two such values.", "---", "### Final Answer", "$$\n\boxed{2}\n$$", "---", "### Bonus Insight", "This straightforward yet elegant model illustrates a key property of absolute value functions: symmetry around $ t = 8 $. Because $ |t - 8| = 4 $ has exactly two solutions equidistant from 8, and the repair rate is defined continuously and symmetrically, we expect two discrete time points—here confirmed to be integer-valued. Future iterations of self-healing materials may involve more complex biology-inspired kinetics, but this simple case reminds us that sometimes clarity and precision deliver the most powerful insights.", "---", "Keywords: self-healing polymer, repair rate, r(t) = √((t−8)²), absolute value function, integer solutions, material science, mm/hour repair, r(t) = 4", "Meta Description:\nHow many integer time values make a self-healing polymer’s repair rate exactly 4 mm/hour? Here’s how solving $ |t - 8| = 4 $ reveals two key moments in the healing process."]