# Maxima and Minima

## Length of one side for maximum area of trapezoid (solution by Calculus)

## 55 - Greatest angle subtended by a picture

**Problem 55**

The lower edge of the picture is a ft, the upper edge is b ft, above the eye of an observer. At what horizontal distance should he stand, if the vertical angle subtended by the picture is to be greatest?

## Problems in Caculus Involving Inverse Trigonometric Functions

The following are problems involving inverse trigonometric functions.

## 26-27 Horizontal rod entering into a room from a perpendicular corridor

**Problem 26**

A corridor 4 ft wide opens into a room 100 ft long and 32 ft wide, at the middle of one side. Find the length of the longest thin rod that can be carried horizontally into the room.

## 24-25 Largest rectangle inscribed in a circular quadrant

**Problem 24**

Find the area of the largest rectangle that can be cut from a circular quadrant as in Fig. 76.

## 23 - Sphere cut into a pyramid

**Problem 23**

A sphere is cut in the form of a right pyramid with a square base. How much of the material can be saved?

## 22 - Smallest cone that may circumscribe a sphere

**Problem 22**

A sphere of radius *a* is dropped into a conical vessel full of water. Find the altitude of the smallest cone that will permit the sphere to be entirely submerged.

## 20-21 Width of the second corridor for a pole to pass horizontally

**Problem 20**

A pole 24 feet long is carried horizontally along a corridor 8 feet wide and into a second corridor at right angles to the first. How wide must the second corridor be?

**Problem 21**

Solve Problem 20 if the pole is of length $L$ and the first corridor is of width $C$.

## 19 Direction of the man to reach his destination as soon as possible

**Problem 19**

A man on an island *a* miles south of a straight beach wishes to reach a point on shore *b* miles east of his present position. If he can row *r* miles per hour and walk *w* miles per hour, in what direction should he row, to reach his destination as soon as possible? See Fig. 57.

## 17-18 A man in a motorboat needs to catch a bus

**Problem 17**

A man in a motorboat at A receives a message at noon, calling him to B. A bus making 40 miles per hour leaves C, bound for B, at 1:00 PM. If AC = 30 miles, what must be the speed of the boat, to enable the man to catch the bus?