Respuesta :
The height and radius of the cone-shaped cup that will use the smallest amount of paper, that is the maximum volume is 3.72 and 2.632 respectively.
Formula used in the solution is
Volume of the cone= (1/3)π(r^2)h
Area of the cone= πr√(l^2 + r^2)
Given, the volume of the cone is 27 cm^3.
(1/3)π(r^2)h =27
Simplifying the equation to get the value of r.
π(r^2)=81/h
r^2=81/hπ
[tex]r=\sqrt{\frac{81}{\pi h} }[/tex]
Substituting the value of r in area, we get
[tex]A=\pi \sqrt{\frac{81}{\pi h} } \sqrt{h^{2} + \frac{81}{\pi h} }[/tex]
[tex]\frac{dA}{dh} =\pi \sqrt{\frac{81}{\pi h} } \sqrt{h^{2} + \frac{81}{\pi h} }[/tex]
dA/dh=√81 × (π - 162/[tex]h^{3}[/tex]) × 1/√(πh + 81/[tex]h^{2}[/tex])
At dA/dh=0, we will get,
√81 × (π - 162/[tex]h^{3}[/tex]) × 1/√(πh + 81/[tex]h^{2}[/tex])=0
√81 × (π - 162/[tex]h^{3}[/tex])=0
[tex]\pi - \frac{162}{h^{3} }=0[/tex]
Thus, [tex]\pi h^{3} -162=0[/tex]
[tex]\pi h^{3} =162[/tex]
[tex]h^{3} =\frac{162}{\pi}[/tex]
h^3= 51.5662015618
h= 3.7221
Now, substitute h in radius,
[tex]r=\sqrt{\frac{81}{\pi \times 3.7221} }[/tex]
r=2.632
Hence, the height of the cone shaped cup is 3.7221 and the radius is 2.632.
To learn more on cone here:
https://brainly.com/question/23863102#
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