91Ó°ÊÓ

The orbit of the earth around the sun is elliptical in shape with the sun at one focus, a semimajor axis of length \(92.9\) million miles, and an eccentricity of \(0.017\). Find (a) how close the earth gets to the sun and (b) the greatest possible distance between the earth and the sun.

The cost of production of a commodity is $$\$ 12$$ less per unit at a point \(A\) than it is at a point \(B\) and the distance between \(A\) and \(B\) is 100 miles. Assuming that the route of delivery of the commodity is along a straight line, and that the delivery cost is 20 cents per unit per mile, find the curve at any point of which the commodity can be supplied from either \(A\) or \(B\) at the same total cost. (HINT: Take points \(A\) and \(B\) at \((-50,0)\) and \((50,0)\), respectively.)

A tank has a horizontal axis of length \(20 \mathrm{ft}\) and its ends are semiellipses. The width across the top of the tank is \(10 \mathrm{ft}\) and the depth is \(6 \mathrm{ft}\). If the tank is full of water, how much work is necessary to pump all the water to the top of the tank?

If a parabola has its focus at the origin and the \(x\) axis is its axis, prove that it must have an equation of the form \(y^{2}=4 k x+4 k^{2}, k \neq 0\).