Question正十二面体の頂点 $\mathrm{O}, \mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}$ に関する問題です。 (1) 点 $\mathrm{O}$ と平面 $\mathrm{P}$ の距離を証明せよ。 (2) 点 $\mathrm{O}$ を含む立体の体積を求めよ。

Studdy Solution

STEP 1

Assumptions1. The figure is a regular dodecahedron with each side length of. . The vertices O, A, B, C, D are on the dodecahedron.
3. The four points A, B, C, D are on the same plane, denoted as.
4. The length of the line segment AB is $1+\sqrt{5}$ , which can be used without proof.
5. We need to prove that the distance between point O and plane is1.
6. We need to find the volume of the solid that includes point O when the regular dodecahedron is cut into two solids by plane.

STEP 2

First, we need to find the height of the regular dodecahedron. The height of a regular dodecahedron with side length a is given by the formula $h = a \sqrt{2(5+ \sqrt{5})}$

STEP 3

Now, plug in the given value for the side length to calculate the height.
$h =2 \sqrt{2(5+ \sqrt{5})}$

STEP 4

Calculate the height of the regular dodecahedron.
$h =2 \sqrt{2(+ \sqrt{})} =2 \sqrt{20+2\sqrt{20}} =2 \sqrt{10+ \sqrt{400}} =2 \sqrt{10+20} =2 \sqrt{30} =2 \times.48 =10.96$

STEP 5

The distance between point O and plane is half of the height of the regular dodecahedron.
$d = \frac{h}{2}$

STEP 6

Plug in the value for the height to calculate the distance.
$d = \frac{10.96}{2}$

STEP 7

Calculate the distance between point O and plane.
$d = \frac{10.96}{2} =5.48$

STEP 8

Now, we need to find the volume of the regular dodecahedron. The volume of a regular dodecahedron with side length a is given by the formula $V = \frac{15+7\sqrt{5}}{4}a^3$

STEP 9

Now, plug in the given value for the side length to calculate the volume.
$V = \frac{15+7\sqrt{5}}{4} \times2^3$

STEP 10

Calculate the volume of the regular dodecahedron.
$V = \frac{15+7\sqrt{5}}{4} \times2^3 =61.3$

STEP 11

The volume of the solid that includes point O is half of the volume of the regular dodecahedron.
$V = \frac{V}{}$

STEP 12

Plug in the value for the volume to calculate the volume of the solid.
$V = \frac{61.}{2}$

STEP 13

Calculate the volume of the solid that includes point O.
$V = \frac{61.3}{2} =30.65$ The distance between point O and plane is5.48 and the volume of the solid that includes point O is30.65.

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Question正十二面体の頂点 $\mathrm{O}, \mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}$ に関する問題です。 (1) 点 $\mathrm{O}$ と平面 $\mathrm{P}$ の距離を証明せよ。 (2) 点 $\mathrm{O}$ を含む立体の体積を求めよ。

Studdy Solution

STEP 1

STEP 2

First, we need to find the height of the regular dodecahedron. The height of a regular dodecahedron with side length a is given by the formula $h = a \sqrt{2(5+ \sqrt{5})}$

STEP 3

Now, plug in the given value for the side length to calculate the height.
$h =2 \sqrt{2(5+ \sqrt{5})}$

STEP 4

Calculate the height of the regular dodecahedron.
$h =2 \sqrt{2(+ \sqrt{})} =2 \sqrt{20+2\sqrt{20}} =2 \sqrt{10+ \sqrt{400}} =2 \sqrt{10+20} =2 \sqrt{30} =2 \times.48 =10.96$

STEP 5

The distance between point O and plane is half of the height of the regular dodecahedron.
$d = \frac{h}{2}$

STEP 6

Plug in the value for the height to calculate the distance.
$d = \frac{10.96}{2}$

STEP 7

Calculate the distance between point O and plane.
$d = \frac{10.96}{2} =5.48$

STEP 8

Now, we need to find the volume of the regular dodecahedron. The volume of a regular dodecahedron with side length a is given by the formula $V = \frac{15+7\sqrt{5}}{4}a^3$

STEP 9

Now, plug in the given value for the side length to calculate the volume.
$V = \frac{15+7\sqrt{5}}{4} \times2^3$

STEP 10

Calculate the volume of the regular dodecahedron.
$V = \frac{15+7\sqrt{5}}{4} \times2^3 =61.3$

STEP 11

The volume of the solid that includes point O is half of the volume of the regular dodecahedron.
$V = \frac{V}{}$

STEP 12

Plug in the value for the volume to calculate the volume of the solid.
$V = \frac{61.}{2}$

STEP 13

Calculate the volume of the solid that includes point O.
$V = \frac{61.3}{2} =30.65$ The distance between point O and plane is5.48 and the volume of the solid that includes point O is30.65.

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Question正十二面体の頂点 O,A,B,C,D\mathrm{O}, \mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}O,A,B,C,D に関する問題です。 (1) 点 O\mathrm{O}O と平面 P\mathrm{P}P の距離を証明せよ。 (2) 点 O\mathrm{O}O を含む立体の体積を求めよ。

Studdy Solution

STEP 1

STEP 2

First, we need to find the height of the regular dodecahedron. The height of a regular dodecahedron with side length a is given by the formulah=a2(5+5)h = a \sqrt{2(5+ \sqrt{5})}h=a2(5+5​)​

STEP 3

Now, plug in the given value for the side length to calculate the height.h=22(5+5)h =2 \sqrt{2(5+ \sqrt{5})}h=22(5+5​)​

STEP 4

Calculate the height of the regular dodecahedron.h=22(+)=220+220=210+400=210+20=230=2×.48=10.96h =2 \sqrt{2(+ \sqrt{})} =2 \sqrt{20+2\sqrt{20}} =2 \sqrt{10+ \sqrt{400}} =2 \sqrt{10+20} =2 \sqrt{30} =2 \times.48 =10.96h=22(+​)​=220+220​​=210+400​​=210+20​=230​=2×.48=10.96

STEP 5

The distance between point O and plane is half of the height of the regular dodecahedron.d=h2d = \frac{h}{2}d=2h​

STEP 6

Plug in the value for the height to calculate the distance.d=10.962d = \frac{10.96}{2}d=210.96​

STEP 7

Calculate the distance between point O and plane.d=10.962=5.48d = \frac{10.96}{2} =5.48d=210.96​=5.48

STEP 8

Now, we need to find the volume of the regular dodecahedron. The volume of a regular dodecahedron with side length a is given by the formulaV=15+754a3V = \frac{15+7\sqrt{5}}{4}a^3V=415+75​​a3

STEP 9

Now, plug in the given value for the side length to calculate the volume.V=15+754×23V = \frac{15+7\sqrt{5}}{4} \times2^3V=415+75​​×23

STEP 10

Calculate the volume of the regular dodecahedron.V=15+754×23=61.3V = \frac{15+7\sqrt{5}}{4} \times2^3 =61.3V=415+75​​×23=61.3

STEP 11

The volume of the solid that includes point O is half of the volume of the regular dodecahedron.V=VV = \frac{V}{}V=V​

STEP 12

Plug in the value for the volume to calculate the volume of the solid.V=61.2V = \frac{61.}{2}V=261.​

STEP 13

Calculate the volume of the solid that includes point O.V=61.32=30.65V = \frac{61.3}{2} =30.65V=261.3​=30.65The distance between point O and plane is5.48 and the volume of the solid that includes point O is30.65.

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Question正十二面体の頂点 $\mathrm{O}, \mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}$ に関する問題です。 (1) 点 $\mathrm{O}$ と平面 $\mathrm{P}$ の距離を証明せよ。 (2) 点 $\mathrm{O}$ を含む立体の体積を求めよ。

First, we need to find the height of the regular dodecahedron. The height of a regular dodecahedron with side length a is given by the formula $h = a \sqrt{2(5+ \sqrt{5})}$

Now, plug in the given value for the side length to calculate the height.
$h =2 \sqrt{2(5+ \sqrt{5})}$

Calculate the height of the regular dodecahedron.
$h =2 \sqrt{2(+ \sqrt{})} =2 \sqrt{20+2\sqrt{20}} =2 \sqrt{10+ \sqrt{400}} =2 \sqrt{10+20} =2 \sqrt{30} =2 \times.48 =10.96$

The distance between point O and plane is half of the height of the regular dodecahedron.
$d = \frac{h}{2}$

Plug in the value for the height to calculate the distance.
$d = \frac{10.96}{2}$

Calculate the distance between point O and plane.
$d = \frac{10.96}{2} =5.48$

Now, we need to find the volume of the regular dodecahedron. The volume of a regular dodecahedron with side length a is given by the formula $V = \frac{15+7\sqrt{5}}{4}a^3$

Now, plug in the given value for the side length to calculate the volume.
$V = \frac{15+7\sqrt{5}}{4} \times2^3$

Calculate the volume of the regular dodecahedron.
$V = \frac{15+7\sqrt{5}}{4} \times2^3 =61.3$

The volume of the solid that includes point O is half of the volume of the regular dodecahedron.
$V = \frac{V}{}$

Plug in the value for the volume to calculate the volume of the solid.
$V = \frac{61.}{2}$

Calculate the volume of the solid that includes point O.
$V = \frac{61.3}{2} =30.65$ The distance between point O and plane is5.48 and the volume of the solid that includes point O is30.65.