Question

By how much does the volume of a cylinder change if the radius is halved and the height is doubled? a) The volume is quartered. d) The volume doubles. b) The volume is cut in half. e) The volume quadruples. c) There is no change in the volume.

Short answer

Question: When the radius of a cylinder is halved and its height is doubled, the resulting cylinder's volume is: a) The same as the original b) Cut in half c) Doubled d) Quadrupled Answer: b) Cut in half

Step-by-step solution

Write the formula for the volume of a cylinder

The volume (V) of a cylinder can be calculated using the formula: V = π * r^2 * h where r is the radius of the cylinder and h is its height.

Apply the changes to the radius and height

We are given that the radius is halved and the height is doubled. So, the new radius (r') will be (1/2)r and the new height (h') will be 2h. Now, let's find the new volume (V') of the cylinder with these changes: V' = π * (r')^2 * h'

Substitute the new radius and height into the formula

Substitute the new values of the radius and height into the volume formula: V' = π * ((1/2)r)^2 * (2h)

Calculate the new volume

Simplify the equation to find the new volume (V'): V' = π * (1/4)r^2 * (2h) = (1/2)(π * r^2 * h)

Compare the new volume with the original volume

Comparing the new volume (V') with the original volume (V), we can see that the new volume is half of the original volume: V' = (1/2)V Hence, the volume of the cylinder changes by a factor of 1/2. So, the correct answer is: b) The volume is cut in half.

Key concepts

Cylinder Volume Formula

Understanding the volume of a cylinder is a fundamental concept in geometry that involves calculating the amount of space inside a cylindrical object. The volume formula for a cylinder, expressed as V = \( \pi r^2 h \), provides the key to unlocking this measurement. Here, \( r \) stands for the radius of the circular base, while \( h \) implies the height of the cylinder, and \( \pi \) is the constant approximately equal to 3.14159.

This formula gives rise to many practical applications, ranging from determining the amount of water a pipe can hold, to calculating fuel volumes in tankers. In order to use this formula effectively, it's important to grasp the concept of squares and multiplication within geometric spaces.

Geometry of Cylindrical Shapes

Geometry is the branch of mathematics that deals with the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. A cylinder is a solid geometric figure with straight parallel sides and a circular or oval section. It's one of the fundamental geometric shapes in three dimensions.

Key Characteristics of a Cylinder

In terms of geometry, a cylinder is defined by its radius and height. The radius of the cylinder correlates to the size of its base, and the height determines the extent of its parallel sides. In addition to these, the concept of symmetry plays a vital role in cylindrical shape geometry. Cylinders exhibit rotational symmetry around their central axis.

Understanding these geometric properties is essential when dealing with problems that involve calculating the volumes or surface areas of cylindrical objects.

Volumetric Alteration in Cylinders

Volumetric alteration refers to the change in volume of a three-dimensional object when it undergoes modifications to its dimensions. When working with cylinders, any modifications to the radius or height directly affect the overall volume.

Example of Volumetric Alteration

In the provided exercise, the radius is halved, and the height is doubled. Applying these changes, the new volume V' = \( \pi (\frac{1}{2}r)^2 \cdot 2h \) simplifies to \( \frac{1}{2}\pi r^2 h \), which is half the original volume. So, the change caused by differing the radius and height results in a diminished volume, specifically by a factor of 1/2.

Understanding how these dimensions interact to influence volume is critical in various fields, from industrial design to, more abstractly, understanding the implications of scale in geometric shapes.