Question

A certain wood stove has a firebox volume of \(4.25 \mathrm{ft}^{3} .\) What firebox volume would be expected if all dimensions of the stove were increased by a factor of \(1.25 ?\)

Short answer

The expected firebox volume with dimensions increased by a factor of 1.25 is 8.32 ft^3.

Step-by-step solution

Understand the Scaling Factor

Identify the scaling factor applied to all dimensions of the stove. In this case, the scaling factor is 1.25, which means each dimension (length, width, height) of the stove is increased by a factor of 1.25.

Calculate the Volume Scaling Factor

Since volume is a three-dimensional measurement, the volume scaling factor will be the cube of the linear scaling factor. Therefore, calculate the volume scaling factor by cubing the scaling factor 1.25.

Multiply the Original Volume by the Volume Scaling Factor

Multiply the original volume of the wood stove, which is 4.25 ft^3, by the volume scaling factor obtained in Step 2 to find the expected firebox volume after all dimensions have been increased.

Key concepts

Scaling Factor

Understanding the scaling factor is crucial when it comes to resizing objects while maintaining their proportionality. A scaling factor is a number by which all dimensions of an object are multiplied to result in a scaled up or scaled down version of the object. When an object's dimensions are increased by the same scaling factor, the object grows uniformly in all directions. If a wood stove has a scaling factor of 1.25 as per our exercise, it means that each dimension—be it length, width, or height—is multiplied by 1.25.

Consider a simple example where a square of side 2 units is scaled by a factor of 1.5. The new side length would be 2 * 1.5, resulting in 3 units. This uniform growth ensures that the scaled object retains the same shape as the original. The concept of the scaling factor is not confined to any specific number of dimensions; it applies just as fluidly in two-dimensional scaling (such as images on a screen) as it does in three-dimensional scaling (like our wood stove).

Geometric Scaling in Mathematics

Geometric scaling in mathematics refers to the process of enlarging or reducing a figure or a three-dimensional object in a proportional manner. This concept is used extensively in various fields such as architecture, engineering, and design to create models or resize existing structures.

The key to geometric scaling is the consistent application of the scaling factor to all relevant measurements. For instance, scaling a two-dimensional figure requires multiplying both its length and width by the scaling factor, while scaling a three-dimensional object likewise involves all three dimensions—length, width, and height. An important property of geometric scaling is that while the shape of the object remains the same, its area (in the case of two-dimensional figures) or volume (in the case of three-dimensional objects) changes according to specific mathematical relationships. Notably, when dealing with volume, the relationship involves cubing the scaling factor.

Cubic Volume Calculation

Cubic volume calculation is the process of determining the three-dimensional space occupied by an object, which is typically measured in cubic units. For a rectangular object, the volume is calculated by multiplying its length by its width by its height. When it comes to scaling the cubic volume, things become more interesting.

If the scaling factor for an object's dimensions is known, the volume scaling factor is found by raising this scaling factor to the third power (cubing it), given that volume is a function of three dimensions. This change occurs because volume is calculated by multiplying three linear dimensions together. Therefore, increasing each linear dimension by a scaling factor increases the volume by the scale factor cubed. In our example involving the wood stove with an original volume of 4.25 ft³ and a linear scaling factor of 1.25, the volume scaling factor would be calculated as: \(1.25^3 = 1.953125\). The new volume would then be found by multiplying the original volume (4.25 ft³) by this volume scaling factor, providing us with the scaled-up volume of the wood stove's firebox.