Set \( C(t) = 10t e^{-0.1t} \), then \( C' = 10e^{-0.1t}(1 - 0.1t) \), set to -1:

["# Understanding the Function ( C(t) = 10t e^{-0.1t} ) and Its Derivative — Solving for When ( C'(t) = -1 )", "In mathematical modeling, differential equations and functions involving exponential decay are ubiquitous across physics, economics, and engineering. One particularly useful function is ( C(t) = 10t e^{-0.1t} ), which models growth modulated by exponential decay — common in phenomena like cooling, radioactive decay with feedback, or production rates subject to tapering.", "This article explores the function ( C(t) = 10t e^{-0.1t} ), derives its derivative ( C'(t) ), and solves the equation ( C'(t) = -1 ), offering insight into when this decreasing phase begins and how steep it becomes.", "---", "## What Is the Function ( C(t) = 10t e^{-0.1t} )?", "The function ( C(t) = 10t e^{-0.1t} ) combines a linear term ( 10t ) with an exponential decay ( e^{-0.1t} ). This product forms a product rule function, prevalent in calculus and applied math.", "### Key Features:\n- Initial Behavior: At ( t = 0 ), ( C(0) = 0 ).\n- Growth then Decay: Initially increases due to the linear term, but modeled decay ( e^{-0.1t} ) causes eventual decline.\n- Applications: Real-world systems such as population growth with resource limits, sales lifespans with saturation, or investment returns with friction.", "---", "## Derivative of ( C(t) )", "To analyze how ( C(t) ) changes over time, compute its derivative using the product rule:", "[\nC(t) = 10t \cdot e^{-0.1t}\n]", "[\nC'(t) = \frac{d}{dt}[10t] \cdot e^{-0.1t} + 10t \cdot \frac{d}{dt}[e^{-0.1t}]\n]", "[\nC'(t) = 10e^{-0.1t} + 10t (-0.1)e^{-0.1t}\n]", "[\nC'(t) = 10e^{-0.1t} - t e^{-0.1t}\n]", "Factor out ( 10e^{-0.1t} ):", "[\nC'(t) = 10e^{-0.1t}(1 - 0.1t)\n]", "This elegant expression reveals how the rate of change depends on both the time ( t ) and the exponential dampening factor.", "---", "## Finding When ( C'(t) = -1 )", "We solve:", "[\n10e^{-0.1t}(1 - 0.1t) = -1\n]", "Divide both sides by 10:", "[\ne^{-0.1t}(1 - 0.1t) = -0.1\n]", "Let ( x = -0.1t ), so ( t = -10x ). Substituting:", "[\ne^{x}(1 + x) = -0.1\n]", "Now solve:", "[\n(1 + x)e^{x} = -0.1\n]", "This transcendental equation cannot be solved algebraically. Instead, use numerical methods or a graphing approach to find the root.", "---", "### Numerical Solution", "Consider ( f(x) = (1 + x)e^{x} + 0.1 ). We seek ( x ) where ( f(x) = 0 ).", "- At ( x = -1 ): ( (0)e^{-1} + 0.1 = 0.1 > 0 )\n- At ( x = -1.2 ): ( (-0.2)e^{-1.2} + 0.1 \approx -0.2(0.3012) + 0.1 = -0.06024 + 0.1 = 0.03976 > 0 )\n- At ( x = -1.3 ): ( (-0.3)e^{-1.3} + 0.1 \approx -0.3(0.2725) + 0.1 = -0.08175 + 0.1 = 0.01825 > 0 )\n- At ( x = -1.4 ): ( (-0.4)e^{-1.4} + 0.1 \approx -0.4(0.2466) + 0.1 = -0.09864 + 0.1 = 0.00136 > 0 )\n- At ( x = -1.41 ): ( (-0.41)e^{-1.41} + 0.1 \approx -0.41(0.2433) + 0.1 \approx -0.09897 + 0.1 = 0.00103 )\n- At ( x = -1.42 ): ( (-0.42)e^{-1.42} + 0.1 \approx -0.42(0.2416) + 0.1 = -0.1015 + 0.1 = -0.0015 < 0 )", "Root between ( x = -1.41 ) and ( -1.42 ). Interpolating gives approximately ( x \approx -1.415 ).", "Now convert back to ( t ):", "[\nt = -10x \approx -10(-1.415) = 14.15\n]", "Thus, ( C'(t) = -1 ) occurs at approximately ( t \approx 14.15 ).", "---", "## Interpretation: When Does the Rate React from Positive to Negative?", "At ( t \approx 14.15 ), the growth rate ( C'(t) ) reaches ( -1 ), indicating the function transitions from increasing to decreasing. Prior to this, the rate is positive (increasing); after, it becomes negative (decreasing), capturing the tipping point where decay dominates.", "This is crucial for modeling scenarios requiring precise control, optimization, or prediction — such as inventory levels, signal decay, or biological processes with a critical threshold.", "---", "## Conclusion", "The function ( C(t) = 10t e^{-0.1t} ) elegantly captures bounded growth undone by exponential decay. Its derivative,", "[\nC'(t) = 10e^{-0.1t}(1 - 0.1t)\n]", "shows how the slope changes over time. Solving ( C'(t) = -1 ) reveals the pivotal moment ( t \approx 14.15 ), when the function shifts from acceleration to deceleration. Understanding such transitions empowers deeper insight and application in real-world modeling.", "---", "Keywords:\nSet ( C(t) = 10t e^{-0.1t} ), derivative ( C'(t) = 10e^{-0.1t}(1 - 0.1t) ), solve ( C'(t) = -1 ), exponential decay with linear growth, calculus applications, mathematical modeling, real-world growth curves."]