Question

1. geometry the volume v of a regular octahedron with edge length ( ell ) is given by ( v = \frac{ell^3 sqrt{2}}{3} ). write the volume in simplest form for an octahedron with the given edge lengths.

a. ( sqrt{15} ) cm

b. ( sqrt{24} ) cm

c. ( 3sqrt{8} ) cm

Explanation:

Part a: Edge length \( l = \sqrt{15} \) cm

Step 1: Substitute \( l \) into the volume formula

The volume formula for a regular octahedron is \( V=\frac{l^{3}\sqrt{2}}{3} \). Substitute \( l = \sqrt{15} \) into the formula:
\( V=\frac{(\sqrt{15})^{3}\sqrt{2}}{3} \)

Step 2: Simplify \( (\sqrt{15})^{3} \)

We know that \( (\sqrt{a})^{n}=a^{\frac{n}{2}} \), so \( (\sqrt{15})^{3}=15^{\frac{3}{2}} = 15\sqrt{15} \). Then the volume becomes:
\( V=\frac{15\sqrt{15}\times\sqrt{2}}{3} \)

Step 3: Simplify the fraction and the product of square - roots

Simplify \( \frac{15}{3}=5 \), and \( \sqrt{15}\times\sqrt{2}=\sqrt{15\times2}=\sqrt{30} \). So \( V = 5\sqrt{30}\space cm^{3} \)

Part b: Edge length \( l=\sqrt{24} \) cm

Step 1: Substitute \( l \) into the volume formula

Substitute \( l = \sqrt{24} \) into \( V=\frac{l^{3}\sqrt{2}}{3} \):
\( V=\frac{(\sqrt{24})^{3}\sqrt{2}}{3} \)

Step 2: Simplify \( (\sqrt{24})^{3} \)

First, simplify \( \sqrt{24}=\sqrt{4\times6} = 2\sqrt{6} \). Then \( (\sqrt{24})^{3}=(2\sqrt{6})^{3}=2^{3}\times(\sqrt{6})^{3}=8\times6\sqrt{6}=48\sqrt{6} \)
The volume formula becomes \( V=\frac{48\sqrt{6}\times\sqrt{2}}{3} \)

Step 3: Simplify the fraction and the product of square - roots

Simplify \( \frac{48}{3} = 16 \), and \( \sqrt{6}\times\sqrt{2}=\sqrt{6\times2}=\sqrt{12}=\sqrt{4\times3}=2\sqrt{3} \). Then \( V=16\times2\sqrt{3}=32\sqrt{3}\space cm^{3} \)

Part c: Edge length \( l = 3\sqrt{8} \) cm

Step 1: Simplify \( l = 3\sqrt{8} \)

Simplify \( \sqrt{8}=\sqrt{4\times2}=2\sqrt{2} \), so \( l = 3\times2\sqrt{2}=6\sqrt{2} \)

Step 2: Substitute \( l \) into the volume formula

Substitute \( l = 6\sqrt{2} \) into \( V=\frac{l^{3}\sqrt{2}}{3} \):
\( V=\frac{(6\sqrt{2})^{3}\sqrt{2}}{3} \)

Step 3: Simplify \( (6\sqrt{2})^{3} \)

\( (6\sqrt{2})^{3}=6^{3}\times(\sqrt{2})^{3}=216\times2\sqrt{2}=432\sqrt{2} \)
The volume formula becomes \( V=\frac{432\sqrt{2}\times\sqrt{2}}{3} \)

Step 4: Simplify the product of square - roots and the fraction

\( \sqrt{2}\times\sqrt{2} = 2 \), so \( V=\frac{432\times2}{3}=\frac{864}{3}=288\space cm^{3} \)

Answer:

s:
a. \( \boldsymbol{5\sqrt{30}\space cm^{3}} \)
b. \( \boldsymbol{32\sqrt{3}\space cm^{3}} \)
c. \( \boldsymbol{288\space cm^{3}} \)