Concept explainers
Beehives In a beehive each cell is a regular hexagonal prism, as shown in the figure. The amount of wax W in the cell depends on the apex angle θ and is given by
Bees instinctively choose θ so as to use the least amount of wax possible.
- (a) Use a graphing device to graph W as a function of θ for 0 < θ < π.
- (b) For what value of θ does W have its minimum value? [Note: Biologists have discovered that bees rarely deviate from this value by more than a degree or two.]
(a)
To sketch: The graph of
Explanation of Solution
The given function is,
The amount of wax in each cell is represented by W and
Sketch the graph of W with respect to
Figure (1)
Figure (1) shows the graph of
(b)
To find: The value of
Answer to Problem 69E
The amount of wax W is minimum for
Explanation of Solution
In beehive, each cell is regular hexagonal prism, open at one end with a trihedron at the other end.
The amount of wax in each cell is represented by W and
The given relation between W and
From close examination of part (a) it is clear that the lowest part of the curve is at
Thus, for
Chapter 6 Solutions
Precalculus: Mathematics for Calculus - 6th Edition
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- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning