91Ó°ÊÓ

Suppose that a 5 -year projection of population trends suggests that \(t\) years from now, the population of a certain community will be \(P\) thousand, where $$ P(t)=-2 t^{3}+9 t^{2}+8 t+200 $$ a. At what rate will the population be growing 3 years from now? b. At what rate will the rate of population growth be changing with respect to time 3 years from now?

At a certain factory, the daily output is \(Q(L)=20,000 L^{1 / 2}\) units, where \(L\) denotes the size of the labor force measured in worker-hours. Currently 900 worker-hours of labor are used each day. Use calculus to estimate the effect on output that will be produced if the labor force is cut to 885 worker-hours.

Estimate the largest percentage error you can allow in the measurement of the radius of a sphere if you want the error in the calculation of its surface area using the formula \(S=4 \pi r^{2}\) to be no greater than \(8 \%\).

A lantern falls from the top of a building in such a way that after \(t\) seconds, it is \(h(t)=150-16 t^{2}\) feet above ground. A woman 5 feet tall originally standing directly under the lantern sees it start to fall and walks away at the constant rate of \(5 \mathrm{ft} / \mathrm{sec}\). How fast is the length of the woman's shadow changing when the lantern is 10 feet above the ground?

A baseball diamond is a square, 90 feet on a side. A runner runs from second base to third at \(20 \mathrm{ft} / \mathrm{sec}\). How fast is the distance \(s\) between the runner and home plate changing when he is 15 feet from third base?