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Sagot :
The expression for the radius and height of the cone can be obtained from
the property of a function at the maximum point.
- [tex]The \ radius, \ of \ the \ base \ of \ the \ cone \ is \ \sqrt{ \dfrac{3}{4}} \times radius \ of \ circular \ sheet \ metal[/tex]
- The height of the cone is half the length of the radius of the circular sheet metal.
Reasons:
The part used to form the cone = A sector of a circle
The length of the arc of the sector = The perimeter of the circle formed by the base of the cone.
[tex]Volume \ of \ a \ cone = \dfrac{1}{3} \cdot \pi \cdot r^3 \cdot h[/tex]
- [tex]Volume \ of \ a \ cone, \, V = \dfrac{1}{3} \cdot \pi \cdot r^3 \cdot \sqrt{(s^2- r^2)}[/tex]
θ/360·2·π·s = 2·π·r
Where;
s = The radius of he circular sheet metal
h = s² - r²
- [tex]\dfrac{dV}{dr} = \dfrac{d}{dr} \left(\dfrac{1}{3} \cdot \pi \cdot r^3 \cdot \sqrt{(s^2- r^2)}\right) = \dfrac{\pi \cdot (3 \cdot r^2 \cdot s^2 - 4 \cdot r^4)}{\sqrt{(s^2- r^2)}} = 0[/tex]
3·r²·s² - 4·r⁴ = 0
3·r²·s² = 4·r⁴
3·s² = 4·r²
[tex]\underline{\left \right. The \ radius, \, r =\sqrt{ \dfrac{3}{4}} \cdot s}[/tex]
[tex]\underline{The \ height, \, h =\sqrt{s^2 - \dfrac{3}{4}\cdot s^2} = \dfrac{s}{2}}}[/tex]
Learn more here:
https://brainly.com/question/14466080
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