A manager of a dinosaur-themed amusement park orders velociraptors. Their populatio...
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Problem
A manager of a dinosaur-themed amusement park orders velociraptors. Their population is modeled by the function v(t) = 1/5 t^2 - 2t + 12. At what point will the park patrons be safe once again (i.e., there will be no velociraptors left in the park)?
Step-by-step solution
To find when the velociraptor population is zero, we set v(t) = 0 and solve the quadratic equation $1/5 t^2 - 2t + 12 = 0$. Multiplying by 5 gives $t^2 - 10t + 60 = 0$. Using the quadratic formula, $t = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(60)}}{2(1)} = \frac{10 \pm \sqrt{100 - 240}}{2} = \frac{10 \pm \sqrt{-140}}{2}$. Since the discriminant is negative, there are no real solutions, meaning the population never reaches zero.
Answer
14