Question

If a solid block of glass, with a volume of exactly 100 in. \(^{3}\), is placed in a basin of water that is full to the brim, then ______ of water will overflow from the basin.

Short answer

\(100 \: in.^{3}\) of water will overflow from the basin.

Step-by-step solution

Identify the volume of the solid glass block

The problem tells us that the volume of the glass block is exactly 100 cubic inches (in. \(^{3}\)).

Relate the volume of the solid glass block to the volume of the overflowing water

When the glass block is placed in the basin full of water, it causes some water to overflow. The volume of the overflowing water will be equal to the volume of the glass block. In this case, 100 in. \(^{3}\) of water will overflow due to the volume displacement caused by the solid glass block.

Write the answer

100 in. \(^{3}\) of water will overflow from the basin when the glass block is placed in it.

Key concepts

Volume of Solids

Understanding the concept of the volume of solids is essential in many scientific and mathematical contexts. Volume refers to the amount of three-dimensional space occupied by an object. It's particularly useful in determining the capacity of various objects or the quantity of substance they can contain. For most regular-shaped objects, such as cubes, spheres, and cylinders, formulas exist to calculate their volume. For instance, the volume of a cube is found by raising the length of one of its sides to the third power, captured by the equation \( V = a^3 \), where \( V \) represents volume and \( a \) is the length of a side.

In the context of the exercise, we determine the volume of a solid glass block. The problem states that this volume is 100 cubic inches (in. \( ^3 \)). When students understand how to obtain the volume of various shapes, they gain the ability to foresee the impact such objects will have when interacting with fluids, such as water in this case. This realization is foundational for grasping more complex principles like buoyancy or Archimedes' principle.

Density and Buoyancy

Density is a property that links mass and volume by the equation \( \rho = \frac{m}{V} \), where \( \rho \) represents density, \( m \) is mass in grams or kilograms, and \( V \) is volume in cubic centimeters or cubic meters. A solid will float or sink in a fluid depending mostly on how its density compares to the fluid's density.

Buoyancy, on the other hand, is the force that a fluid exerts on an object submerged in it. This buoyant force is equal to the weight of the fluid that the object displaces. An object will float if its overall density is lower than the fluid's, as the buoyant force counteracts gravity. Conversely, it will sink if its density is higher. This idea can be confusing, but a practical example helps illustrate this: ice floats on water because its density is lower than that of liquid water, and so the buoyant force keeps it atop.

To make the concept of buoyancy clearer, let's tie it back to the textbook problem. The glass block, when placed in water, displaces a volume of water equal to its own volume. Here, the fact that the glass is denser than the water is made apparent by it sinking and displacing the water, but we focus solely on the volume displaced to solve the exercise.

Archimedes' Principle

Archimedes' principle is a law of physics fundamental to fluid mechanics. This principle states that the upward buoyant force exerted on a body immersed in a fluid, whether partially or fully submerged, is equal to the weight of the fluid that the body displaces. To put it simply, an object will push away or displace an amount of fluid equivalent to its own volume when it's submerged in the fluid.

The principle can be fascinating when you consider real-life applications. For example, it's what allows gigantic ships made of steel to float in water. Though steel is denser than water, the ship's shape causes it to displace a sufficient volume of water whose weight equals the total weight of the ship, allowing it to float.

In the exercise provided, the Archimedes' principle is directly applied. When the solid glass block with a volume of 100 in. \( ^3 \) is submerged in water, it will displace a corresponding volume of water. That is why 100 in. \( ^3 \) of water will overflow when the block is placed in the full basin: the glass displaces its own volume in water, which has no place but to overflow. This simple demonstration is an excellent way for students to observe Archimedes' principle in action and understand the relationship between volume displacement and buoyant forces.