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Chapter 3: Problem 3

The equation of a right eircular cylinder, whose axis in the \(z\)-axis and radius \(a\) is \((a) x^{2}+y^{2}+z^{2}=a^{2}\) (b) \(a^{2}+y^{2}=a^{2}\) (c) \(x^{2}+y^{2}=a^{2}\) \(\left(\right.\) d) \(2^{2}+x^{2}=a^{2}\)

Short Answer

Expert verified
The correct equation is (c) \(x^2+y^2=a^2\).

Step by step solution

01

Understanding the geometry of a cylinder

A right circular cylinder whose axis is along the \(z\)-axis means that the shapes formed by slicing the cylinder perpendicular to the \(z\)-axis are circles. The cylinder can be described by the equation for a circle in the \(xy\)-plane.
02

Identify the equation for a circle

The equation for a circle with center at the origin \((0, 0)\) in the \(xy\)-plane is \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle. In this case, the radius is given as \(a\).
03

Determine the equation of the cylinder

For a cylinder that extends along the \(z\)-axis, the circular cross-section remains constant across all values of \(z\). Therefore, the equation of the cylinder, unaffected by \(z\), is \(x^2 + y^2 = a^2\).
04

Select the correct option

Upon examining each given choice:(a) \(x^2 + y^2 + z^2 = a^2\) describes a sphere.(b) \(a^2 + y^2 = a^2\) simplifies to \(y = 0\), not a cylinder.(c) \(x^2 + y^2 = a^2\) matches the determined equation.(d) \(2^2 + x^2 = a^2\) doesn't match the required form.Thus, the correct answer is (c) \(x^2+y^2=a^2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

right circular cylinder
A right circular cylinder is a three-dimensional shape that features two parallel circular bases and a curved surface connecting them. These bases are of equal radius and aligned directly opposite one another. It is called 'right' because the sides make a right angle with the bases.
This shape is common and extensively used in both geometry and real-life applications, such as cans and pipes.
  • The radius of the cylinder, typically denoted by "a" or "r", is the distance from the center to the edge of the base.
  • The height or the length of the cylinder is the perpendicular distance between the two bases.
Understanding the properties of a cylinder helps in visualizing its geometry, crucial for solving various mathematical problems that involve this shape.
geometry of a cylinder
The geometry of a cylinder can be broken down into several key properties. These include its height, radius, and surface area, which describe the basic elements.
  • Surface area: Consists of both the lateral surface area plus the area of the two bases. The lateral surface area is evaluated by "unrolling" the curved surface into a rectangle.
  • Volume: Calculated by multiplying the base area by the cylinder height. This captures the capacity of the cylinder.
  • Cross-sections: For a cylinder aligned along a particular axis, its perpendicular cross-sections will always be circular, maintaining the original radius.
These geometrical attributes are at the heart of many applications and form the basis for understanding how cylinders interact with other geometric shapes.
cylinder axis along z-axis
When a cylinder's axis is along the z-axis, its symmetry is developed around this vertical line. This scenario is often used in mathematics due to its alignment with coordinate planes.
With the axis on the z-axis, the problem simplifies to one of planar geometry in the xy-plane.
  • This means any cross-section along a constant z will result in a circle.
  • The equation you derive reflects this simplicity: it is independent of the z-coordinate, shown as: \(x^2 + y^2 = a^2\).
Such configurations are beneficial in simplifying calculations, making them essential in both academic and practical geometric applications.
equation of a sphere
The equation of a sphere is often confused with that of a cylinder, due to their reliance on similar square terms. Specifically, a sphere has the form
\(x^2 + y^2 + z^2 = r^2\), where \(r\) is the radius of the sphere.
This equation denotes that all points on the sphere are equidistant from the center, making it a purely radial symmetry.
  • In contrast to cylinders, spheres use all three coordinate directions in \(x\), \(y\), and \(z\).
  • This distinguishes the sphere from a cylinder, which maintains its shape primarily in two directions and remains unaffected by changes in the third direction.
Understanding the distinction between these two geometrical shapes helps prevent confusion when determining the correct geometric form based on given equations.

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Most popular questions from this chapter

The distanee between the planes \(2 x+2 y+z-6=0\) and \(4 x+4 y+2 z-7=0\) is (c) \(1 / 3\) (b) \(5 / 6\) (c) \(13 / 8\).

The direction cosines of a line which is equally inclined to the coordinute axes are w-

A line malies angles \(o, \beta, \gamma\) with the coordinate axes, then ces \(2 \alpha+\cos 2 \beta+\cos 2 \gamma\) is equal to (a) 1 (b) 2 (c) \(-1\) \((d)-2\)

The equation of the plane pasning through \((4,-2,1)\) and perpendicular to the line with direction nation \(7,2,-3\) in (a) \(x+3 y-4 z-8=0\) (b) \(2 x+7 y-3 z-24=0\) \(\begin{array}{llll}\text { (c) } 7 x+2 y-3 z-21=0 . & \text { (d) } 7 x+3 y-2 z+21 & =0 . & \text { (V.T.t., } 2009 & S\end{array}\)

The line \(x=a y+b, z=c y+d\) and \(x=a^{\prime} y+b^{\prime}, z=c^{\prime} y+d^{\prime}\) are perpendicular if (a) \(a a^{\circ}+c c^{\prime}=1\) (b) \(a a^{\circ}+c c^{\prime}=-1\) (c) \(b b^{2}+d d^{\prime}=1\) (d) \(b b^{\prime}+d d^{\prime}=-1\).
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