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The charcoal from a tree killed in the volcanic eruption that formed Crater Lake in Oregon contained \(44.5 \%\) of the carbon- 14 found in living matter. About how old is Crater Lake?

The equation \(x^{2}=2^{x}\) has three solutions: \(x=2, x=4,\) and one other. Estimate the third solution as accurately as you can by graphing.

A decimal representation of \(e\) Find \(e\) to as many decimal places as your calculator allows by solving the equation \(\ln x=1\) using Newton's method in Section 4.7.

Skydiving If a body of mass \(m\) falling from rest under the action of gravity encounters an air resistance proportional to the square of the velocity, then the body's velocity \(t\) sec into the fall satisfies the differential equation $$m \frac{d v}{d t}=m g-k v^{2}$$ where \(k\) is a constant that depends on the body's aerodynamic properties and the density of the air. (We assume that the fall is short enough so that the variation in the air's density will not affect the outcome significantly.) a. Show that $$ v=\sqrt{\frac{m g}{k}} \tanh (\sqrt{\frac{g k}{m}} t)$$ satisfies the differential equation and the initial condition that \(v=0\) when \(t=0\) b. Find the body's limiting velocity, lim_,-\inftyv. c. For a 160 -lb skydiver \((m g=160),\) with time in seconds and distance in feet, a typical value for \(k\) is \(0.005 .\) What is the diver's limiting velocity?