Respuesta :
[tex]\bf \textit{volume of a cylinder}\\\\
V_c=\pi r^2h\qquad
\begin{cases}
r=radius\\
h=height\\
------\\
r=x\\
h=y
\end{cases}\implies V_c=x^2y\pi
\\\\\\
\textit{volume of a cone}\\\\
V_n=\cfrac{\pi r^2h}{3}\qquad r=2x\implies V_n=\cfrac{\pi \cdot (2x)^2h}{3}
\\\\\\
V_n=\cfrac{\pi \cdot 2^2x^2h}{3}\implies V_n=\cfrac{4x^2h\pi }{3}\\\\
-------------------------------\\\\[/tex]
[tex]\bf V_c=V_n\implies x^2y\pi =\cfrac{4x^2h\pi }{3}\implies 3x^2y\pi =4x^2h\pi \\\\\\ \cfrac{3x^2y\pi }{4x^2\pi }=h\implies \boxed{\cfrac{3y}{4}=h}[/tex]
[tex]\bf V_c=V_n\implies x^2y\pi =\cfrac{4x^2h\pi }{3}\implies 3x^2y\pi =4x^2h\pi \\\\\\ \cfrac{3x^2y\pi }{4x^2\pi }=h\implies \boxed{\cfrac{3y}{4}=h}[/tex]
The height of the cone when its volume is equal to a cylinder of base radius x and height is y, is [tex]\frac{3}{4}y[/tex].
What is the volume of a cylinder?
The volume of a cylinder is the space that is been occupied by the cylinder, it is given by the formula,
[tex]\text{Volume of Cylinder} = \pi r^2 h[/tex]
where r is the radius of the cylinder and h is the height of the cylinder.
What is the volume of the cone?
The volume of the cone is the space occupied by a cone of base radius r and the height h.
[tex]\text{Volume of the cone}=\dfrac{1}{3}\pi r^2h[/tex]
Given to us
The height of the cylinder = y
The radius of the base of the cylinder = x
The radius of the cone base = 2x
Let the height of the cone be represented by h.
As it is given to us that the volume of the cone and the cylinder is same, therefore,
[tex]\text{Volume of cylinder}= \text{Volume of the cone}[/tex]
Substitute the values,
[tex]\pi (x)^2y = \dfrac{1}{3} \pi (2x)^2 h\\\\y = \dfrac{1}{3} 4 h\\\\h = \dfrac{3y}{4}[/tex]
Hence, the height of the cone when its volume is equal to a cylinder of base radius x and height is y, is [tex]\frac{3}{4}y[/tex].
Learn more about Volume of Cone:
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