Fostering Advanced Algebraic Thinking with Casio Technology
Sample Solution—Homemade Waffle Cones (continued)
c.
Again, without measuring, write an equation for the height of the cone you
constructed in terms of the arc length you removed from the circle.
The height of the cone can be determined using the Pythagorean Theorem, as
the radius of the base of the cone and the height of the cone are two legs of a
right triangle in which the slant height is the hypotenuse. Therefore, (radius of
cone)
2
+ (height of cone)
2
= (slant height of cone)
2
, or (height of cone)
2
=
(slant height of cone)
2
(radius of cone)
2
. Substituting the expressions from
parts a and b:
(height of cone)
2
= (10.5)
2
2
2
510
¸
¹
·
¨
©
§
S
a
.
(height of cone)
2
= (10.5)
2
»
»
¼
º
«
«
¬
ª
¸
¹
·
¨
©
§
2
2
2
510
510
SS
a.
.
(height of cone)
2
=
2
2
4
510
S
S
aa.
height of cone =
2
2
4
510
S
S
aa.
Note: we are only considering the positive root.
d.
Write an equation for the volume of the cone in terms of the arc length you
removed.
Recall, the volume of a cone is V =
hr
2
3
1
S
. Using the expressions from parts b
and c, the volume of the cone we constructed is:
V =
2
2
2
4
510
2
510
3
1
S
SS
S
aa.a
.
¸
¹
·
¨
©
§
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