Respuesta :
Answer:
Radius = 22.04 unit,
Height = 15.58 unit
Step-by-step explanation:
Since, the volume of cone,
[tex]V=\frac{1}{3}\pi x^2 h[/tex]
Where,
x = radius,
h = height,
Since, the slant height of a cone is,
[tex]l=\sqrt{x^2 + h^2}[/tex]
[tex]l^2 = x^2 + h^2[/tex]
[tex]\implies h=\sqrt{l^2 - x^2}[/tex]
Here, l = 27.
[tex]\implies h = \sqrt{27^2 - x^2}=\sqrt{729 - x^2}----(1)[/tex]
So, the volume would be,
[tex]V=\frac{1}{3}\pi x^2 (\sqrt{729 - x^2})[/tex]
Differentiating with respect to x,
[tex]\frac{dV}{dt}=\frac{1}{3}\pi (x^2 \times \frac{1}{2\sqrt{729 - x^2}}\times -2x + \sqrt{729 - x^2}\times 2x)[/tex]
[tex]=\frac{1}{3}\pi (\frac{-x^3+2x(729-x^2)}{\sqrt{729 - x^2}}[/tex]
[tex]=\frac{1}{3}\pi (\frac{-x^3+1458x-2x^3}{\sqrt{729 - x^2}})[/tex]
[tex]=\frac{1}{3}\pi (\frac{-3x^3+1458x}{\sqrt{729 - x^2}})[/tex]
[tex]=\pi (\frac{-x^3+486x}{\sqrt{729 - x^2}})[/tex]
For maxima or minima,
[tex]\frac{dV}{dt}=0[/tex]
[tex]\implies -x^3 + 486x = 0[/tex]
[tex]x^3 = 486x[/tex]
[tex]x^2 = 486[/tex]
[tex]\implies x=\pm \sqrt{486}=\pm 22.04[/tex]
But side can not be negative,
So, x = 22.04,
Since,
[tex]\frac{dV}{dx}|_{x=20}=297.91\text{Positive}[/tex]
While,
[tex]\frac{dV}{dx}|_{x=23}=-219.70\text{Negative}[/tex]
Thus, by the first derivative test,
V(x) is maximum at x = 22.04,
Also, from equation (1),
h = 15.58
Hence, for maximising the volume, the radius and height of the cone would be 22.04 unit and 15.58 unit respectively.
