Question:
The volume of the tetrahedron bounded by the planes \(x=1\), \(y=2\), \(z=3\) and \(12x+8y+6z=70\) is rounded off to one decimal place.
The volume of the tetrahedron bounded by the planes \(x=1\), \(y=2\), \(z=3\) and \(12x+8y+6z=70\) is rounded off to one decimal place.
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For a tetrahedron cut from coordinate planes by \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\), the volume is \(\frac{abc}{6}\).
Updated On: Jun 1, 2026
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Correct Answer: 4
Solution and Explanation
Step 1: Shift the coordinate system.
Let
\[ X=x-1,\qquad Y=y-2,\qquad Z=z-3 \]
Then the planes \(x=1\), \(y=2\), and \(z=3\) become
\[ X=0,\qquad Y=0,\qquad Z=0 \]
Step 2: Substitute in the fourth plane.
Given plane is
\[ 12x+8y+6z=70 \]
Substitute
\[ x=X+1,\quad y=Y+2,\quad z=Z+3 \]
Step 3: Simplify the plane equation.
\[ 12(X+1)+8(Y+2)+6(Z+3)=70 \]
\[ 12X+12+8Y+16+6Z+18=70 \]
\[ 12X+8Y+6Z+46=70 \]
\[ 12X+8Y+6Z=24 \]
Step 4: Convert into intercept form.
\[ \frac{X}{2}+\frac{Y}{3}+\frac{Z}{4}=1 \]
So the intercepts on the \(X\), \(Y\), and \(Z\) axes are
\[ 2,\quad 3,\quad 4 \]
Step 5: Use tetrahedron volume formula.
If intercepts are \(a\), \(b\), and \(c\), then volume is
\[ V=\frac{1}{6}abc \]
Step 6: Substitute values.
\[ V=\frac{1}{6}\times 2\times 3\times 4 \]
\[ V=\frac{24}{6} \]
Step 7: Final answer.
\[ V=4 \]
Rounded off to one decimal place,
\[ \boxed{4.0} \]
Let
\[ X=x-1,\qquad Y=y-2,\qquad Z=z-3 \]
Then the planes \(x=1\), \(y=2\), and \(z=3\) become
\[ X=0,\qquad Y=0,\qquad Z=0 \]
Step 2: Substitute in the fourth plane.
Given plane is
\[ 12x+8y+6z=70 \]
Substitute
\[ x=X+1,\quad y=Y+2,\quad z=Z+3 \]
Step 3: Simplify the plane equation.
\[ 12(X+1)+8(Y+2)+6(Z+3)=70 \]
\[ 12X+12+8Y+16+6Z+18=70 \]
\[ 12X+8Y+6Z+46=70 \]
\[ 12X+8Y+6Z=24 \]
Step 4: Convert into intercept form.
\[ \frac{X}{2}+\frac{Y}{3}+\frac{Z}{4}=1 \]
So the intercepts on the \(X\), \(Y\), and \(Z\) axes are
\[ 2,\quad 3,\quad 4 \]
Step 5: Use tetrahedron volume formula.
If intercepts are \(a\), \(b\), and \(c\), then volume is
\[ V=\frac{1}{6}abc \]
Step 6: Substitute values.
\[ V=\frac{1}{6}\times 2\times 3\times 4 \]
\[ V=\frac{24}{6} \]
Step 7: Final answer.
\[ V=4 \]
Rounded off to one decimal place,
\[ \boxed{4.0} \]
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