Question

The dimensions of a prism are doubled. a. How does the surface area change? b. How does the volume change?

Short answer

The surface area quadruples and the volume octuples.

Step-by-step solution

Understand Initial Formulas

For a rectangular prism, the surface area is calculated by the formula: \( SA = 2lw + 2lh + 2wh \) where \( l \), \( w \), and \( h \) are the length, width, and height, respectively. The volume is given by \( V = lwh \).

Understand the Effect of Doubling Dimensions

Doubling the dimensions means replacing \( l \), \( w \), and \( h \) with \( 2l \), \( 2w \), and \( 2h \).

Calculate New Surface Area

Substitute the doubled dimensions into the surface area formula: \( SA_{new} = 2(2l)(2w) + 2(2l)(2h) + 2(2w)(2h) = 8lw + 8lh + 8wh = 4(2lw + 2lh + 2wh) = 4(SA) \). The new surface area is four times the original surface area.

Calculate New Volume

Substitute the doubled dimensions into the volume formula: \( V_{new} = (2l)(2w)(2h) = 8lwh = 8V \). The new volume is eight times the original volume.

Key concepts

Surface Area

The surface area of a rectangular prism is an important concept in geometry, and it's vital to understand how it's calculated. Imagine a rectangular prism as a box with six faces. To find the surface area, we need to calculate the total area of all the faces combined. The formula for the surface area of a rectangular prism is given by:\[ SA = 2lw + 2lh + 2wh \]where:

• \( l \) is the length
• \( w \) is the width
• \( h \) is the height

To see how the surface area changes when dimensions are altered, it's crucial to apply any changes to all three variables. If we double the length, width, and height, our new dimensions are \( 2l \), \( 2w \), and \( 2h \). Substituting these values into our original formula, we calculate:\[ SA_{new} = 2(2l)(2w) + 2(2l)(2h) + 2(2w)(2h) \]This simplifies to:\[ SA_{new} = 8lw + 8lh + 8wh = 4(SA) \]Thus, the new surface area becomes four times the original surface area. This is an important pattern to note: if each dimension of the prism doubles, the surface area increases by a factor of four.

Volume

Volume is a fundamental concept in geometry that measures the amount of space a three-dimensional object occupies. For a rectangular prism, the volume can be found using the formula:\[ V = lwh \]where \( l \), \( w \), and \( h \) are the prism's length, width, and height. Visualize this formula as filling the box with unit cubes, each having a volume of 1 cubic unit.
When we double the dimensions of our rectangular prism, we replace \( l \), \( w \), and \( h \) with \( 2l \), \( 2w \), and \( 2h \). By substituting these into the volume formula, it becomes:\[ V_{new} = (2l)(2w)(2h) \]Simplifying this, we find:\[ V_{new} = 8lwh = 8V \]This indicates the new volume is eight times the original volume. Doubling each dimension results in an exponential increase in volume compared to the linear change in surface area. This is because volume is a function of three dimensions, and each increase multiplies the total result by two, leading to an overall multiplicative factor of eight.

Rectangular Prism

Rectangular prisms are ubiquitous geometric shapes commonly referred to as boxes or rectangular cuboids. They are part of the broader category known as polyhedra, specifically prisms, characterized by two parallel and congruent bases and flat rectangular faces. In a rectangular prism, each face is a rectangle, and all angles are right angles, contributing to its straightforward properties.
A rectangular prism's defining feature is its six faces, 12 edges, and 8 vertices. These features can be visualized by imagining a common shoebox. The area of each face contributes to the overall surface area, while the interior space determines its volume. These calculations rely on simple mathematical operations involving length, width, and height.

• The surface area is determined by summing up the area of all rectangles
• Volume identifies the number of cubic units that can fit inside

Understanding these basic computations helps solve more complex geometry problems while also providing practical solutions for real-world applications like packaging, storage, and construction. Recognizing how dimensions affect both surface area and volume can prove essential for various academic and professional settings.