A tent is to be constructed from a 20' x 20' square piece of canvas by cutting out symmetrical regions from each of its sides as indicated in the figure. Then the four remaining triangular flaps are to be turned up and stitched together to form the shape of a pyramid with a square base. (a) In the figure, find the exact lengths |OB| and |BA| in terms of x. Do not use decimal approximations in your answers. (b) Given the well known fact that the volume of any pyramid is: V = (area of base) × (height) then show that the volume of the tent is: 3 V = (2 · |OB|)² × √|AB|² — |OB|² i.e., V = ²x² × √√200 – 20x . (c) Use calculus to find the value of x that maximizes the volume of the pyramid.
A tent is to be constructed from a 20' x 20' square piece of canvas by cutting out symmetrical regions from each of its sides as indicated in the figure. Then the four remaining triangular flaps are to be turned up and stitched together to form the shape of a pyramid with a square base. (a) In the figure, find the exact lengths |OB| and |BA| in terms of x. Do not use decimal approximations in your answers. (b) Given the well known fact that the volume of any pyramid is: V = (area of base) × (height) then show that the volume of the tent is: 3 V = (2 · |OB|)² × √|AB|² — |OB|² i.e., V = ²x² × √√200 – 20x . (c) Use calculus to find the value of x that maximizes the volume of the pyramid.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:A tent is to be constructed from a 20' x 20' square piece
of canvas by cutting out symmetrical regions from each
of its sides as indicated in the figure. Then the four
remaining triangular flaps are to be turned up and
stitched together to form the shape of a pyramid with a
square base.
(a) In the figure, find the exact lengths |OB| and |BA|
in terms of x. Do not use decimal approximations
in your answers.
(b) Given the well known fact that the volume of any
pyramid is:
V = (area of base) × (height)
then show that the volume of the tent is:
V = ½ (2 · |OB|)² × √|AB|² — |OB|²
i.e.,
V = ²x² × √√200 20x
(c) Use calculus to find the value of x that
maximizes the volume of the pyramid.
fold up
B
X
O
20'
T
B
X
201
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