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1. The fission reactor of a nuclear plant explodes at $t=0$. The following function $d$ represents the density in radioactive particles per metre cubed in the air inside the plant, as a function of $t$ in hours: \[ d(t)=t e^{10-t} . \] It is deemed safe to enter the nuclear plant as soon as $d$ is below 4 . (a) Suppose we used Newton's Method with an initial guess of $t_{1}=10$ to find an approximation for the time when the density of the radioactive particles equals zero. Would Newton's Method ever find a time when the radioactive particle density equals zero? Explain. (b) Use L'Hospital's Rule to show that the density of radioactive particles tends to zero in the long run. (c) Compute the third degree Taylor polynomial, $T_{3}(t)$, of the function $d(t)$ at $t=10$. (d) Use $T_{3}(t)$ to approximate the value of $d\left(10+\frac{b}{6}\right)$. (Round your answer to two decimal places.) Assuming that this approximation is good enough, is it safe to enter the nuclear plant after $\left(10+\frac{b}{6}\right)$ hours? Explain.

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