04-05 Stiffness and strength of timber beam
Problem 4
The stiffness of a rectangular beam is proportional to the breadth and the cube of the depth. Find the shape of the stiffest beam that can be cut from a log of a given size.
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03 - Heaviest cylinder that can be made from a shot
Problem 3
Find the weight of the heaviest circular cylinder can be cut from a 16-lb shot.
02 - Cylinder of maximum convex area inscribed in a sphere
Problem 02
A cylinder is inscribed in a given sphere. Find the shape of the cylinder if its convex surface area is a maximum.
01 Rectangle of maximum perimeter inscribed in a circle
Problem 01
Find the shape of the rectangle of maximum perimeter inscribed in a circle.
Maxima and Minima Using Trigonometric Functions
Many problems in application of maxima and minima may be solved easily by making use of trigonometric functions. The basic idea is the same; identify the constant terms and identify the variable to be maximized or minimized, differentiate that variable then equate to zero.
Chapter 4 - Trigonometric and Inverse Trigonometric Functions
Differentiation of Trigonometric Functions
Trigonometric identities and formulas are basic requirements for this section. If u is a function of x, then
37-38 How fast a ship leaving from its starting point
Problem 37
A ship sails east 20 miles and then turns N 30° W. If the ship's speed is 10 mi/hr, find how fast it will be leaving the starting point 6 hr after the start.
35-36 Time Rates: Lengthening of shadow and movement of its tip in 3D space
Problem 35
An arc light hangs at the height of 30 ft above the center of a street 60 ft wide. A man 6 ft tall walks along the sidewalk at the rate of 4 ft/sec. How fast is his shadow lengthening when he is 40 ft up the street?
33-34 Time Rates: A car traveling east and airplane traveling north
Problem 33
From a car traveling east at 40 miles per hour, an airplane traveling horizontally north at 100 miles per hour is visible 1 mile east, 2 miles south, and 2 miles up. Find when this two will be nearest together.
31-32 Train in an elevated track and car in perpendicular road
Problem 31
An elevated train on a track 30 ft above the ground crosses a street at the rate of 20 ft/sec at the instant that a car, approaching at the rate of 30 ft/sec, is 40 ft up the street. Find how fast the train and the car separating 1 second later.