91Ó°ÊÓ

Maximizing Volume Find the dimensions of a right circular cylinder of maximum volume that can be inscribed in a sphere of radius 10 cm. What is the maximum volume?

Maximizing Profit Suppose \(r(x)=8 \sqrt{x}\) represents revenue and \(c(x)=2 x^{2}\) represents cost, with \(x\) measured in thousands of units. Is there a production level that maximizes profit? If so, what is it?

Particle Motion A particle moves along the parabola \(y=x^{2}\) in the first quadrant in such a way that its \(x\) -coordinate (in meters) increases at a constant rate of 10 \(\mathrm{m} / \mathrm{sec} .\) How fast is the angle of inclination \(\theta\) of theline joining the particle to the origin changing when \(x=3 ?\)

Minimizing Average cost Suppose \(c(x)=x^{3}-10 x^{2}-30 x\) where \(x\) is measured in thousands of units. Is there a production level that minimizes average cost? If so, what is it?

Minimizing Average cost Suppose \(c(x)=x e^{x}-2 x^{2},\) where \(x\) is measured in thousands of units. Is there a production level that minimizes average cost? If so, what is it?

In Exercises \(25-28,\) a particle is moving along the \(x\) -axis with position function \(x(t) .\) Find the (a) velocity and (b) acceleration, and (c) describe the motion of the particle for \(t \geq 0\) . $$x(t)=6-2 t-t^{2}$$

Particle Motion A particle moves from right to left along the parabolic curve \(y=\sqrt{-x}\) in such a way that its \(x\) -coordinate (in meters) decreases at the rate of 8 \(\mathrm{m} / \mathrm{sec} .\) How fast is the angle of inclination \(\theta\) of the line joining the particle to the origin changing when $x=-4 ?

Melting Ice A spherical iron ball is coated with a layer of ice of uniform thickness. If the ice melts at the rate of 8 \(\mathrm{mL} / \mathrm{min}\) , how fast is the outer surface area of ice decreasing when the outer diameter (ball plus ice) is 20 \(\mathrm{cm} ? \)

In Exercises \(25-28,\) a particle is moving along the \(x\) -axis with position function \(x(t) .\) Find the (a) velocity and (b) acceleration, and (c) describe the motion of the particle for \(t \geq 0\) . $$x(t)=t^{3}-3 t+3$$

Cubic Polynomial Functions Let $$f(x)=a x^{3}+b x^{2}+c x+d, a \neq 0$$ (a) Show that f has either 0 or 2 local extrema. (b) Give an example of each possibility in part (a).