## 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?

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

Problem
BC of trapezoid ABCD is tangent at any point on circular arc DE whose center is O. Find the length of BC so that the area of ABCD is maximum.

Solution
As described by Alexander Bogomolny of cut-the-knot.org, for maximum area of trapezoid, the point of tangency should be at the midline of AB and DC, thus H is the midpoint of BC.

## 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.

Problem 27
Solve Problem 26 if the room is 56 feet long.

## 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.

Problem 25
In Problem 24, draw the graph of A as a function of $\theta$. Indicating the portion of the curve that has a meaning.

## 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?

Problem 18
Solve Problem 17, if AC = 20 miles and the bus makes 50 miles per hour, leaving C at 12:18 PM, bound for B.