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$\backslash$table$[[$Entered$,$Answer Preview,Result$],[0\mathrm{.}000124,0\mathrm{.}000124,$incorrect$],[758,758,$correct$]]$

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$(1$ point$)$ The radioactive isotope carbon$-14$ is present in small quantities in all life forms, and it is constantly replenished until the organism dies, after which it decays to stable carbon$-12$ at a rate proportional to the amount of carbon$-14$ present, with a half$-$life of $5568$ years. Suppose $C(t)$ is the amount of carbon$-14$ present at time $t.$

$($a$)$ Find the value of the constant $k$ in the differential equation $C^{\text{'}}=-kC.$

$k=$

$($b$)$ In $1988$ three teams of scientists found that the Shroud of Turin, which was reputed to be the burial cloth of Jesus, contained about $91$ percent of the amount of carbon$-14$ contained in freshly made cloth of the same material$[1].$ How was old the Shroud of Turin in $1988,$ according to these data?

Age $=$ years

$[1]$: The New York Times, October $18,1988.$