Visualization of extreme values with GeoGebra
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Task:
The perimeter of an isosceles triangle is 24cm (Figure 1). A body in 3D is made with the
rotation of the triangle around its base. How much should be the sides of the triangle, so
that the 3D body has maximum volume?
Figure 1
Solution:
With the rotation of an isosceles triangle around the base (Figure 1) a body of two
cones with a common base with radius r and height b / 2 is created (Figure 2).
Figure 2
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Biljana Janakievska
1
Visualization of extreme values with GeoGebra
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In the condition of the task the perimeter is 24sm.
So 2 24. Hence 12 (*)
From figure 2 we see that (**)
The volume of the rotating body is a sum of the volume of the two of the cones
which are the same:
2 2 We'll present the volume of the rotating body as a function of the base of the triangle. By
replacing the values of a from (*) and r from (**) we get:
!
"12 #
$
!
%= 144 12 412 Now, we want to find the maximum of the function.
We find:
&
'412 (& 48 8 86 &
0 т.е. 6 0, we get 6см.
From From 12 we get 9см.
&&
'86 (& =81 1 0
Now we look for the second derivative: So, when the isosceles triangle has a base b = 6sm and side a = 9sm the rotating body
which is obtained by rotation of an isosceles triangle around the base has a maximum
volume.
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Biljana Janakievska
2