Question

The cylinder and the cone have the same volume. What is the height of the cone?

Answer 1

Answer:
`h = (3)/(4)y` is the cone's height.

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Work Shown:

Let's set up each volume equation.

`\text{VL} = \text{volume of cylinder}\\\\\text{VL} = \pi*r^2*h\\\\\text{VL} = \pi*x^2*y\\\\`

`\text{VC} = \text{volume of cone}\\\\\text{VC} = (1)/(3)*\pi*r^2*h\\\\\text{VC} = (1)/(3)*\pi*(2x)^2*h\\\\\text{VC} = (4)/(3)\pi x^2*h\\\\`

Then we're told that VL and VC are the same value. Let's equate the right hand sides and solve for h.

`\text{VL} = \text{VC}\\\\\pi x^2y = (4)/(3)\pi x^2 h\\\\x^2y = (4)/(3)x^2 h\\\\3x^2y = 4x^2 h\\\\h = (3x^2y)/(4x^2)\\\\h = (3y)/(4)\\\\h = (3)/(4)y\\\\`

Answer 2

3y

Explanation:

`\boxed{\begin{minipage}{4 cm}\underline{Volume of a cylinder}\\\\$V=\pi r^2 h$\\\\where:\\ \phantom{ww}$\bullet$ $(r)$ is the radius. \\ \phantom{ww}$\bullet$ $h$ is the height.\\\end{minipage}}`
`\boxed{\begin{minipage}{4 cm}\underline{Volume of a cone}\\\\$V=(1)/(3) \pi r^2 h$\\\\where:\\ \phantom{ww}$\bullet$ $(r)$ is the radius. \\ \phantom{ww}$\bullet$ $h$ is the height.\\\end{minipage}}`

Given dimensions of the cylinder:

• r = x
• h = y

Therefore, the equation for the volume of the cylinder is:

`\implies V_(\sf cylinder)=\pi x^2y`

Given dimensions of the cone:

• diameter of base = 2x ⇒ r = x
• h = ?

Therefore, the equation for the volume of the cone is:

`\implies V_(\sf cone)=(1)/(3) \pi x^2 h`

As the cylinder and the cone have the same volume, substitute the equation for the volume of the cylinder into the equation for the volume of the cone an solve for h:

`\begin{aligned}V_(\sf cylinder)&=V_(\sf cone)\\\implies \pi x^2 y&=(1)/(3) \pi x^2 h\\y&=(1)/(3) h\\3y&=h\\h&=3y\end{aligned}`

Therefore, the height of the cone is 3y.