91Ó°ÊÓ

Find the equation of the tangent to the ellipse \(\left(x^{2} / 16\right)+\left(y^{2} / 9\right)=1\) at \(x\) \(=\sqrt{7}\). There are two answers. Ans. \(y \pm \frac{9}{4}=(x-\sqrt{7})\left(\pm \frac{\sqrt{7}}{4}\right)\).

(a) Find the rate at which the area of a circle changes with respect to time in terms of the time rate of change of the radius. Ans. \(d A / d t=2 \pi r d r / d t\). (b) If, when the radius of a circle is 5 feet, it is increasing at the rate of \(\frac{1}{2} \mathrm{ft} / \mathrm{sec}\), at what rate is the area changing? Ans. \(5 \pi \mathrm{sq} \mathrm{ft} / \mathrm{sec}\). (c) Since when the radius is 5 , it is changing at the rate of \(\frac{1}{2} \mathrm{ft} / \mathrm{sec}\), and the area is then changing at the rate of \(5 \pi \mathrm{sq} \mathrm{ft} / \mathrm{sec}\), does the area increase by \(5 \pi \mathrm{sq} \mathrm{ft}\) in the next second? (d) Suppose that the radius' rate of increase of \(\frac{1}{2} \mathrm{ft} / \mathrm{sec}\) is constant, that is, the same at all values of \(r\). Does the area increase by \(5 \pi\) sq \(\mathrm{ft}\) in the next second after the radius is \(5 \mathrm{ft}\) ? Ans. No.

Two ships start at the same point, but the first ship leaves at noon and sails east at the rate of \(20 \mathrm{mi} / \mathrm{hr}\) and the second leaves at 1 P.M. and sails south at \(25 \mathrm{mi} / \mathrm{hr}\). How fast is the distance between them changing at 2 P.M.?

The orbit of the earth about the sun is practically an ellipse with the sun at one focus. The ratio of the earth's least distance from the sun (which occurs when the earth is at one end of the major axis) to the greatest distance (which occurs at the other end of the major axis) is \(29 / 30\). Find the eccentricity of the ellipse. Ans. \(1 / 59\).

In the case of an ellipse \(a\) is always greater than (or at least equal to) \(b\). What is the corresponding relation of \(a\) to \(b\) for the hyperbola?