1. 10 m C 0 10 m A large conical tank is to be formed from a circular piece of sheet metal of radius 10 meters by cutting out a sector with vertex angle and then welding together the straight edges of the remaining piece. In this problem, you will ultimately determine the value of the vertex angle of the sector that is to be removed in order to maximize the tank's volume. Join & weld these edges together. Figure 2 10m h Figure 1 (a) It is evident that the slant height of any conical tank, so constructed, must necessarily be 10 meters. Express the volume, V, of any cone with slant height 10m as a function of only its altitude, h. Hint: Apply the Pythagorean Theorem to relate r to h (see Figure 2) and recall: V Cone = (Base Area) × (height). (b) Use your result from (a) and the properties of the derivative to find the altitude, h, of the conical tank having slant height 10m and maximal volume. Also, specify the radius, r, and the circumference of the base, CBases of this maximal conical tank. (c) Recall the formula for arc length from geometry: s=re where is measured in radians. Use this formula to find the lengths: S and C from Figure 1, in terms of only the angle 0. (d) Observe that the arc length, C, in Figure 1 corresponds with the circumference of the conical tank, CBaser in Figure 2. Now set your formula for C from (c) equal to the known circumference, CBaser of the maximal conical tank from (b) and solve for 0. Convert your answer for 0 to degrees.
1. 10 m C 0 10 m A large conical tank is to be formed from a circular piece of sheet metal of radius 10 meters by cutting out a sector with vertex angle and then welding together the straight edges of the remaining piece. In this problem, you will ultimately determine the value of the vertex angle of the sector that is to be removed in order to maximize the tank's volume. Join & weld these edges together. Figure 2 10m h Figure 1 (a) It is evident that the slant height of any conical tank, so constructed, must necessarily be 10 meters. Express the volume, V, of any cone with slant height 10m as a function of only its altitude, h. Hint: Apply the Pythagorean Theorem to relate r to h (see Figure 2) and recall: V Cone = (Base Area) × (height). (b) Use your result from (a) and the properties of the derivative to find the altitude, h, of the conical tank having slant height 10m and maximal volume. Also, specify the radius, r, and the circumference of the base, CBases of this maximal conical tank. (c) Recall the formula for arc length from geometry: s=re where is measured in radians. Use this formula to find the lengths: S and C from Figure 1, in terms of only the angle 0. (d) Observe that the arc length, C, in Figure 1 corresponds with the circumference of the conical tank, CBaser in Figure 2. Now set your formula for C from (c) equal to the known circumference, CBaser of the maximal conical tank from (b) and solve for 0. Convert your answer for 0 to degrees.
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:1.
10 m
C
0
10 m
A large conical tank is to be formed from a
circular piece of sheet metal of radius 10
meters by cutting out a sector with vertex
angle and then welding together the
straight edges of the remaining piece. In this
problem, you will ultimately determine the
value of the vertex angle of the sector
that is to be removed in order to maximize
the tank's volume.
Join & weld these edges together.
Figure 2
10m
h
Figure 1
(a) It is evident that the slant height of any conical tank, so constructed, must necessarily be
10 meters. Express the volume, V, of any cone with slant height 10m as a function of
only its altitude, h. Hint: Apply the Pythagorean Theorem to relate r to h (see Figure 2)
and recall: V Cone = (Base Area) × (height).
(b) Use your result from (a) and the properties of the derivative to find the altitude, h, of the
conical tank having slant height 10m and maximal volume. Also, specify the radius, r, and
the circumference of the base, CBases of this maximal conical tank.
(c) Recall the formula for arc length from geometry: s=re where is measured in radians.
Use this formula to find the lengths: S and C from Figure 1, in terms of only the angle 0.
(d) Observe that the arc length, C, in Figure 1 corresponds with the circumference of the
conical tank, CBaser in Figure 2. Now set your formula for C from (c) equal to the known
circumference, CBaser of the maximal conical tank from (b) and solve for 0. Convert your
answer for 0 to degrees.
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