Bubble gum's density is about \(1 \mathrm{g} / \mathrm{cm}^{3} .\) You blow an 8 -g wad of gum into a bubble \(10 \mathrm{cm}\) in diameter. What's the bubble's thickness? (Hint: Think about spreading the bubble into a flat sheet. The surface area of a sphere is \(4 \pi r^{2} .\) )

Understanding the surface area of a sphere is crucial when dealing with problems involving spherical objects, such as a bubble. The surface area is the total area covered by the exterior surface of the sphere. The mathematical formula used to calculate this is given by
\(A = 4 \pi r^2\)
where \(A\) is the surface area and \(r\) is the radius of the sphere. To visualize this, imagine the sphere as being composed of many tiny squares, where the surface area is the sum of the area of these squares. In the case of a bubble, calculating the surface area allows us to understand how much material is needed to cover the bubble's outer layer entirely.

Applying the Formula to a Bubble

As seen in the given problem, for a bubble with a diameter of 10 cm, the radius \(r\) would be half of that, which is 5 cm. Using the formula, the surface area of the bubble would be \(A = 4 \pi (5 \text{ cm})^2 = 100 \pi \text{ cm}^2\). This calculation is the first step in determining the bubble's thickness.

An important concept in physical science is the volume to mass ratio, which is quite simply the amount of space an object occupies (volume) compared to its mass. This ratio is pivotal in many scientific and engineering calculations, as it allows for the determination of an object's density. It helps in understanding how heavy an object is relative to its size.
In formulas, the volume to mass ratio is the inverse of density and is expressed as:
\( \text{Volume to Mass Ratio} = \frac{Volume}{Mass}\)
In the context of the bubble problem, we know the mass of the gum used to create the bubble, which is 8 grams. Given that the density of bubble gum is 1 g/cm³, we can calculate the volume by rearranging the density formula: \(Volume = \frac{Mass}{Density} \).

Density's Role in Calculation

Knowing that density is the mass per unit volume, we can see that the bubble gum's density of 1 g/cm³ means that 1 cm³ of bubble gum has a mass of 1 gram. By dividing the mass of the bubble gum (8g) by its density (1 g/cm³), we find the volume of gum, which is 8 cm³. It's this relationship that informs us how much space the bubble gum takes before it’s inflated into a bubble.

Density is a key concept that describes how compact or concentrated a material is. It is defined as the mass per unit volume of a substance and generally expressed in units such as g/cm³ or kg/m³. Informally, it tells us how much 'stuff' is packed into a certain space.
The formula for density is given as:
\(Density = \frac{Mass}{Volume}\)
In the world around us, density explains why some objects float while others sink, and it’s a crucial factor in material science, engineering, and environmental studies. For example, a material with high density will have more mass in the same volume compared to a material with low density.

Impact on the Bubble Thickness

The density of the bubble gum is given as 1 g/cm³, indicative of bubble gum being relatively dense. In practical terms, this means that a 1 cm³ volume of this bubble gum will weigh 1 gram. This information is essential in calculating the bubble’s thickness because it relates to the amount of gum material that makes up the bubble's volume. The volume, combined with the bubble's surface area, is what ultimately allows us to solve for the bubble's thickness, providing key insights into the properties of the bubble.