Question

The area of the curved surface
\(S = \left\{ (x, y, z) ∈ \R^3 : z^2 = (x − 1)^2 + (y − 2)^2 \right\}\)
lying between the planes z = 2 and z = 3 is

• A. \(4\pi\sqrt2\)
• B. \(5\pi\sqrt2\)
• C. \(9\pi\)
• D. \(9\pi\sqrt2\)

Answer 1

The Correct Option is B

The given problem asks for the area of the curved surface defined by the equation \( S = \left\{ (x, y, z) ∈ \R^3 : z^2 = (x − 1)^2 + (y − 2)^2 \right\} \) between the planes \(z = 2\) and \(z = 3\). Let's solve this step by step.

Step 1: Understand the Equation of the Surface

The given equation \(z^2 = (x - 1)^2 + (y - 2)^2\) represents a cone in 3D space. The vertex of this cone is at (1, 2, 0) and the axis is parallel to the z-axis.

Step 2: Identify the Circle of Intersection with Planes

To find the curved surface area between z = 2 and z = 3, we need to determine the radii of the circles where the cone intersects these planes.

• For z = 2, substituting into the cone's equation gives \( 4 = (x - 1)^2 + (y - 2)^2 \). The radius of this circle is 2.
• For z = 3, substituting gives \( 9 = (x - 1)^2 + (y - 2)^2 \). The radius of this circle is 3.

Step 3: Surface Area of a Cone Section

The area of the curved surface of a cone (frustum) can be calculated using the formula:

\( A = \pi (R_1 + R_2) \sqrt{(R_2 - R_1)^2 + h^2} \)

where \(R_1\) and \(R_2\) are the radii of the circles, and \(h\) is the vertical height between the two planes. Here, \(h = 3 - 2 = 1\).

Step 4: Calculate the Curved Surface Area

Substitute the values into the formula:

\( A = \pi (2 + 3) \sqrt{(3 - 2)^2 + 1^2} \)

\( A = \pi \times 5 \times \sqrt{1 + 1} = 5\pi\sqrt{2} \)

Conclusion

The area of the curved surface of the cone between the planes \(z = 2\) and \(z = 3\) is \(5\pi\sqrt2\), which matches the correct answer.