QuestionFind the volume between a cone and a cube, with the cone's base inscribed in the cube's face. Use edge length $s$ .

Studdy Solution

STEP 1

Assumptions1. The base of the cone is inscribed in one face of the cube, which means the diameter of the base of the cone is equal to the edge length of the cube. . The vertex of the cone is in the opposite face of the cube, which means the height of the cone is equal to the edge length of the cube.
3. The volume of a cube is given by the formula $V_{cube} = s^3$ , where $s$ is the length of the edge of the cube.
4. The volume of a cone is given by the formula $V_{cone} = \frac{1}{3}\pi r^ h$ , where $r$ is the radius of the base of the cone and $h$ is the height of the cone.

STEP 2

First, let's express the radius of the base of the cone in terms of the edge length of the cube. Since the diameter of the base of the cone is equal to the edge length of the cube, the radius of the base of the cone is half the edge length of the cube.
$r = \frac{s}{2}$

STEP 3

Now, let's express the height of the cone in terms of the edge length of the cube. Since the vertex of the cone is in the opposite face of the cube, the height of the cone is equal to the edge length of the cube.
$h = s$

STEP 4

Next, let's calculate the volume of the cube using the formula $V_{cube} = s^3$ .

STEP 5

Now, let's calculate the volume of the cone using the formula $V_{cone} = \frac{1}{3}\pi r^2 h$ and the expressions for $r$ and $h$ that we found in steps2 and3.
$V_{cone} = \frac{1}{3}\pi \left(\frac{s}{2}\right)^2 s$

STEP 6

implify the expression for the volume of the cone.
$V_{cone} = \frac{1}{3}\pi \left(\frac{s^2}{4}\right) s = \frac{1}{12}\pi s^3$

STEP 7

The volume of the region between the cone and the cube is the difference between the volume of the cube and the volume of the cone.
$V_{region} = V_{cube} - V_{cone}$

STEP 8

Substitute the expressions for $V_{cube}$ and $V_{cone}$ that we found in steps4 and6 into the equation from step7.
$V_{region} = s^3 - \frac{1}{12}\pi s^3$

STEP 9

Factor out $s^3$ from the right side of the equation.
$V_{region} = s^3 \left( - \frac{}{12}\pi\right)$ The volume of the region between the cone and the cube is a function of the length of the edge of the cube and is given by $V_{region} = s^3 \left( - \frac{}{12}\pi\right)$ .

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QuestionFind the volume between a cone and a cube, with the cone's base inscribed in the cube's face. Use edge length $s$ .

Studdy Solution

STEP 1

STEP 2

First, let's express the radius of the base of the cone in terms of the edge length of the cube. Since the diameter of the base of the cone is equal to the edge length of the cube, the radius of the base of the cone is half the edge length of the cube.
$r = \frac{s}{2}$

STEP 3

Now, let's express the height of the cone in terms of the edge length of the cube. Since the vertex of the cone is in the opposite face of the cube, the height of the cone is equal to the edge length of the cube.
$h = s$

STEP 4

Next, let's calculate the volume of the cube using the formula $V_{cube} = s^3$ .

STEP 5

Now, let's calculate the volume of the cone using the formula $V_{cone} = \frac{1}{3}\pi r^2 h$ and the expressions for $r$ and $h$ that we found in steps2 and3.
$V_{cone} = \frac{1}{3}\pi \left(\frac{s}{2}\right)^2 s$

STEP 6

implify the expression for the volume of the cone.
$V_{cone} = \frac{1}{3}\pi \left(\frac{s^2}{4}\right) s = \frac{1}{12}\pi s^3$

STEP 7

The volume of the region between the cone and the cube is the difference between the volume of the cube and the volume of the cone.
$V_{region} = V_{cube} - V_{cone}$

STEP 8

Substitute the expressions for $V_{cube}$ and $V_{cone}$ that we found in steps4 and6 into the equation from step7.
$V_{region} = s^3 - \frac{1}{12}\pi s^3$

STEP 9

Factor out $s^3$ from the right side of the equation.
$V_{region} = s^3 \left( - \frac{}{12}\pi\right)$ The volume of the region between the cone and the cube is a function of the length of the edge of the cube and is given by $V_{region} = s^3 \left( - \frac{}{12}\pi\right)$ .

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QuestionFind the volume between a cone and a cube, with the cone's base inscribed in the cube's face. Use edge length sss.

Studdy Solution

STEP 1

STEP 2

First, let's express the radius of the base of the cone in terms of the edge length of the cube. Since the diameter of the base of the cone is equal to the edge length of the cube, the radius of the base of the cone is half the edge length of the cube.r=s2r = \frac{s}{2}r=2s​

STEP 3

Now, let's express the height of the cone in terms of the edge length of the cube. Since the vertex of the cone is in the opposite face of the cube, the height of the cone is equal to the edge length of the cube.h=sh = sh=s

STEP 4

Next, let's calculate the volume of the cube using the formula Vcube=s3V_{cube} = s^3Vcube​=s3.

STEP 5

Now, let's calculate the volume of the cone using the formula Vcone=13πr2hV_{cone} = \frac{1}{3}\pi r^2 hVcone​=31​πr2h and the expressions for rrr and hhh that we found in steps2 and3.Vcone=13π(s2)2sV_{cone} = \frac{1}{3}\pi \left(\frac{s}{2}\right)^2 sVcone​=31​π(2s​)2s

STEP 6

implify the expression for the volume of the cone.Vcone=13π(s24)s=112πs3V_{cone} = \frac{1}{3}\pi \left(\frac{s^2}{4}\right) s = \frac{1}{12}\pi s^3Vcone​=31​π(4s2​)s=121​πs3

STEP 7

The volume of the region between the cone and the cube is the difference between the volume of the cube and the volume of the cone.Vregion=Vcube−VconeV_{region} = V_{cube} - V_{cone}Vregion​=Vcube​−Vcone​

STEP 8

Substitute the expressions for VcubeV_{cube}Vcube​ and VconeV_{cone}Vcone​ that we found in steps4 and6 into the equation from step7.Vregion=s3−112πs3V_{region} = s^3 - \frac{1}{12}\pi s^3Vregion​=s3−121​πs3

STEP 9

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QuestionFind the volume between a cone and a cube, with the cone's base inscribed in the cube's face. Use edge length $s$ .

First, let's express the radius of the base of the cone in terms of the edge length of the cube. Since the diameter of the base of the cone is equal to the edge length of the cube, the radius of the base of the cone is half the edge length of the cube.
$r = \frac{s}{2}$

Now, let's express the height of the cone in terms of the edge length of the cube. Since the vertex of the cone is in the opposite face of the cube, the height of the cone is equal to the edge length of the cube.
$h = s$

Next, let's calculate the volume of the cube using the formula $V_{cube} = s^3$ .

Now, let's calculate the volume of the cone using the formula $V_{cone} = \frac{1}{3}\pi r^2 h$ and the expressions for $r$ and $h$ that we found in steps2 and3.
$V_{cone} = \frac{1}{3}\pi \left(\frac{s}{2}\right)^2 s$

implify the expression for the volume of the cone.
$V_{cone} = \frac{1}{3}\pi \left(\frac{s^2}{4}\right) s = \frac{1}{12}\pi s^3$

The volume of the region between the cone and the cube is the difference between the volume of the cube and the volume of the cone.
$V_{region} = V_{cube} - V_{cone}$

Substitute the expressions for $V_{cube}$ and $V_{cone}$ that we found in steps4 and6 into the equation from step7.
$V_{region} = s^3 - \frac{1}{12}\pi s^3$