## 01 Maximum area of triangle of given perimeter

**Problem 1**

Show that the largest triangle of given perimeter is equilateral.

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**Problem 1**

Show that the largest triangle of given perimeter is equilateral.

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**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?

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

**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.

**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.

**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?

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**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.

**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 $C$.

**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.

**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*.