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Solution: Given $ R(4) = 2R(0) $, substitute into the equation: $ 1000e^{4k} = 2 \cdot 1000 $. Simplify to $ e^{4k} = 2 $. Take the natural logarithm: $ 4k = \ln(2) $, so $ k = rac{\ln(2)}{4} $. Thus, $ oxed{\dfrac{\ln(2)}{4}} $.
Question: The self-healing efficiency of a polymer is modeled by $ rac{x^2 - 9}{x - 3} = 5 $. Solve for $ x $.
Solution: Simplify the left side: $ rac{(x-3)(x+3)}{x-3} = x + 3 $, provided $ x

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