91Ó°ÊÓ

Show that the rectangle of maximum area that can be inscribed in a circle of fixed radius \(a\) is a square.

A right circular cylinder is inscribed in a cone of height \(\underline{H}\) and base radius \(R\) so that the axis of the cylinder coincides with the axis of the cone. Determine the dimensions of the cylinder with the largest lateral surface area.

Cells of a Honeycomb The accompanying figure depicts a single prism-shaped cell in a honeycomb. The front end of the prism is a regular hexagon, and the back is formed by the sides of the cell coming together at a point. It can be shown that the surface area of a cell is given by $$ S(\theta)=6 a b+\frac{3}{2} b^{2}\left(\frac{\sqrt{3}-\cos \theta}{\sin \theta}\right) \quad 0<\theta<\frac{\pi}{2} $$ where \(\theta\) is the angle between one of the (three) upper surfaces and the altitude. The lengths of the sides of the hexagon, \(b\), and the altitude, \(a\), are both constants. a. Show that the surface area is minimized if \(\cos \theta=1 / \sqrt{3}\), or \(\theta \approx 54.7^{\circ} .\) (Measurements of actual honeycombs have confirmed that this is, in fact, the angle found in beehives.)b. Using a graphing utility, verify the result of part (a) by finding the absolute minimum of $$ f(\theta)=\frac{\sqrt{3}-\cos \theta}{\sin \theta} \quad 0<\theta<\frac{\pi}{2} $$

Maximizing Profit The total daily profit in dollars realized by the TKK Corporation in the manufacture and sale of \(x\) dozen recordable DVDs is given by the total profit function $$ \begin{aligned} P(x)=-0.000001 x^{3}+0.001 x^{2}+5 x-500 & \\ 0 & \leq x \leq 2000 \end{aligned} $$ Find the level of production that will yield a maximum daily profit.

Electrical Force of a Conductor A ring-shaped conductor of radius \(a\) carrying a total charge \(Q\) induces an electrical force of magnitude $$ F=\frac{Q}{4 \pi \varepsilon_{0}} \cdot \frac{x}{\left(x^{2}+a^{2}\right)^{3 / 2}} $$ where \(\varepsilon_{0}\) is a constant called the permittivity of free space, at a point \(P\), a distance \(x\) from the center, along the line perpendicular to the plane of the ring through its center. Find the value of \(x\) for which \(F\) is greatest.

Poiseuille's Law According to Poiseuille's Law, the velocity (in centimeters per second) of blood \(r \mathrm{~cm}\) from the central axis of an artery is given by $$ v(r)=k\left(R^{2}-r^{2}\right) \quad 0 \leq r \leq R $$ where \(k\) is a constant and \(R\) is the radius of the artery. Show that the flow of blood is fastest along the central axis. Where is the flow of blood slowest?

Show that the graph of the function \(f(x)=x|x|\) has an inflection point at \((0,0)\) but \(f^{\prime \prime}(0)\) does not exist.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(f\) and \(g\) are increasing functions on an interval \(I\), then their product \(f g\) is also increasing on \(I\).

Escape Velocity An object is projected vertically upward fron the earth's surface with an initial velocity \(v_{0}\) of magnitude less than the escape velocity (the velocity that a projectile should have in order to break free of the earth forever). If only the earth's influence is taken into consideration, then the maximum height reached by the rocket is $$ H=\frac{\iota_{0}^{2} R}{2 g R-v_{0}^{2}} $$ where \(R\) is the radius of the earth and \(g\) is the acceleration due to gravity. a. Show that the graph of \(H\) has a vertical asymptote at \(\nu_{0}=\sqrt{2 g R}\), and interpret your result. b. Use the result of part (a) to find the escape velocity. Take the radius of the earth to be \(4000 \mathrm{mi}\left(g=32 \mathrm{ft} / \mathrm{sec}^{2}\right)\). c. Sketch the graph of \(H\) as a function of \(v_{0}\).

Show that a polynomial function of odd degree greater than or equal to three has at least one inflection point.