Wooden Beam A rectangular beam is to be cut from a cylindrical log of diameter 20 cm. The figures show different ways this can be done.
- (a) Express the cross-sectional area of the beam as a function of the angle θ in the figures.
- (b) Graph the function you found in part (a).
- (c) Find the dimensions of the beam with largest cross-sectional area.
(a)
To express: The cross-sectional area of the rectangular beam cut from cylindrical log as a function of
Answer to Problem 65E
The cross-sectional area of the rectangular beam is given by
Explanation of Solution
Given:
The rectangular beam is cut down from a cylindrical log as shown in the figure.
Calculation:
The cross sectional area of the rectangular beam is given by,
Consider the right angle triangle in the rectangular beam with opposite side as depth adjacent side as width and hypotenuse side is
Formula to find depth.
Substitute depth for opposite and
Formula to find width.
Substitute width for adjacent and
Substitute
Here, the cross sectional area is the function of angle
Denote the cross sectional area by function
Thus, the cross sectional area of rectangular beam is given by
(b)
To sketch: The graph of the function
Explanation of Solution
The function for the cross sectional area is given by
The below figure shows the graph of the function
Figure (1)
The graph of the function in Figure (1) oscillates between
(c)
To find: The dimensions of the beam with largest cross sectional area.
Answer to Problem 65E
The dimensions of the beam with the largest cross sectional area of the function
Explanation of Solution
Given:
The cross sectional area of the beam is given by the function
Calculation:
The function
Case (i):
The value of trigonometric function
The function
Case (ii):
The value of trigonometric function
The function
Case (iii):
The value of trigonometric function
The function
Case (iv):
The value of trigonometric function
The function
Case (v):
The value of trigonometric function
The function
Thus, the function
Therefore the width and depth is to be calculated at angle
From part (a),
Substitute
Thus, the dimensions of the beam with the largest cross sectional area of the function
Chapter 6 Solutions
Precalculus: Mathematics for Calculus - 6th Edition
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