Find the dimensions of the largest rectangle which has its upper two vertices on the [math]x[/math]-axis and the other two on the graph of [math]y = x^2 - 27[/math].

Find the dimensions of the rectangle which has its upper two vertices on the [math]x[/math]-axis and the other two on the graph of [math]y = x^2 - 27[/math] and which has maximum perimeter.

A box without a top is to be made by cutting equal squares from the corners of a rectangular piece of tin [math]30[/math] inches by [math]48[/math] inches and bending up the sides. What size should the squares be is the volume of the box is to be a maximum? [ph Hint: If [math]x[/math] is the side of a square, [math]V(x) = x(30-2x)(48-2x)[/math].]

• lab{2.2.14a} A box without a top is to be made by cutting equal squares from the corners of a square piece of tin, [math]18[/math] inches on a side, and bending up the sides. How large should the squares be if the volume of the box is to be as large as possible?
• Generalize \ref{ex2.2.14a} to the largest open-topped box which can be made from a square piece of tin, [math]s[/math] inches on a side.

• Where should a wire [math]20[/math] inches long be cut if one piece is to be bent into a circle, the other piece is to be bent into a square, and the two plane figures are to have areas the sum of which is a maximum?
• Where should the cut be if the sum of areas is to be a minimum?

A man in a canoe is [math]6[/math] miles from the nearest point of the shore of the lake. The shoreline is approximately a straight line and the man wants to reach a point on the shore [math]5[/math] miles from the nearest point. If his rate of paddling is [math]4[/math] miles per hour and he can run [math]5[/math] miles per hour along the shore, where should he land to reach his destination in the shortest possible time?

Prove that the largest isosceles triangle which can be inscribed in a given circle is also equilateral.

Prove that the smallest isosceles triangle which can be circumscribed about a given circle is also equilateral.

What is the smallest positive number that can be written as the sum of two positive numbers [math]x[/math] and [math]y[/math] so that [math]\frac1x + \frac2y = 1[/math]?

An excursion train is to be run for a lodge outing. The railroad company sets the rate at [math]\[/math]10[math] per person if less that [/math]200[math] tickets are sold. They agree to lower the rate per person by [/math]2[math] cents for each ticket sold above the [/math]200[math] mark, but the train will only hold [/math]450<math> people. What number of tickets will give the company the greatest income?