(c) The bead’s height h is labeled in the ?gure. Rewrite your formula from (b) t
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Question
(c) The bead’s height h is labeled in the ?gure. Rewrite your formula from (b) to show that V = ? 6h3.
3. Since your answer in problem 2(c) expresses the volume entirely in terms of h (and not r or R), it means that all beads of the same height have the same volume. In other words, if you started with a sphere the size of an orange and a sphere the size of a basketball and made them each into beads a height of 2 inches, the beads would have the same volume. Explain how this can be true. (Hint: think about the shape of the beads)
4. Do all beads of the same height h also have the same outside surface area (not including the surface area of the cylindrical hole inside)? (Note: you do not need to use an integral to compute the surface area, just discuss it intuitively.)
Explanation / Answer
A bead is a small, decorative object that is formed in a variety of shapes and sizes of a material such as glass, plastic, or wood, and that is pierced for threading or stringing. In our case I assume the bead is in spherical shape with a hole in it.
(a)
The equation of a circle is given by
x^2 + y^2 = R^2
Solving for y, taking just the positive
y = (R^2 - x^2)
The equation of top of rectangular is
y = r
Setting them equal to find the intersection
r = (R^2 - x^2)
r^2 = R^2 - x^2
x^2 = R^2 - r^2
x = (R^2 - r^2) ... taking just the positive
For r=1, R=2
x = (2^2 - 1^2)
= 3
y = 1
Taking double of the integral revolving around x-axis
V = (a to b){[f(x)]^2-[g(x)]^2}dx
= 2*(0 to 3){[ (2^2 - x^2)]^2 - 1^2}dx
= 2*(0 to 3){[(4 - x^2) - 1}dx
= 2*(0 to 3)(3 - x^2)dx
= 2(3x - x^3/3)|0 to 3
= 2(3* 3 - ( 3)^3/3)
= 2(3* 3 - 3)
= 4 3
(b)
Since the figure is not given I assume it is a spherical shape. The volume of a sphere is given by
V = (4/3)r^3
The height h is
h = 2r
r = h/2
Substituting
V = (4/3)(h/2)^3
= (4/3)h^3/2^3
= [4/(3*8)]h^3
= (1/6)h^3
(c)
If the orange has the height h then the volume is
Vo = (1/6)h^3
If the basketball has the height h then the volume is
Vb = (1/6)h^3
If both had the height of 2
Vo = (1/6)2^3
= 4/3
Vb = (1/6)2^3
= 4/3
Hence,
Vo = Vb
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