Questions: Describe in words the region of R^3 represented by the equation(s) or inequalities. y^2 + z^2 ≤ 81

Describe in words the region of R^3 represented by the equation(s) or inequalities.
y^2 + z^2 ≤ 81
Transcript text: Describe in words the region of $\mathbb{R}^{3}$ represented by the equation(s) or inequalities. \[ y^{2}+z^{2} \leq 81 \]
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Solution

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Solution Steps

Step 1: Analyze the given inequality

The inequality provided is: \[ y^{2} + z^{2} \leq 81 \] This represents a relationship between the variables \(y\) and \(z\) in three-dimensional space \(\mathbb{R}^{3}\).

Step 2: Interpret the inequality geometrically

The equation \(y^{2} + z^{2} = 81\) describes a circle in the \(yz\)-plane with a radius of 9 centered at the origin. The inequality \(y^{2} + z^{2} \leq 81\) includes all points inside or on the boundary of this circle.

Step 3: Extend the interpretation to \(\mathbb{R}^{3}\)

In three-dimensional space, the inequality \(y^{2} + z^{2} \leq 81\) represents an infinite cylinder extending along the \(x\)-axis. The cylinder has a radius of 9 and is centered on the \(x\)-axis. This is because the inequality does not involve the \(x\)-coordinate, meaning the region is unbounded in the \(x\)-direction.

Final Answer

The region of \(\mathbb{R}^{3}\) represented by the inequality \(y^{2} + z^{2} \leq 81\) is an infinite cylinder with a radius of 9 centered along the \(x\)-axis.

\[ \boxed{\text{An infinite cylinder with radius 9 centered along the } x\text{-axis.}} \]

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