Question

Calculate the mass of a block of iron $(d=7.86\mathrm{g}/{\mathrm{cm}}^3)$ with dimensions of $52.8\mathrm{cm}\times 6.74\mathrm{cm}\times 3.73\mathrm{cm}$.

Short answer

The mass of the block of iron is approximately $8328.30\mathrm{g}$.

Step-by-step solution

Calculate the Volume

We begin by calculating the volume of the iron block. The volume of a cuboid is given by the formula Volume = length * breadth * height. Plugging in the given dimensions, we get $Volume=52.8\mathrm{cm}\times 6.74\mathrm{cm}\times 3.73\mathrm{cm}$.

Compute the Volume

After multiplying these values, the volume comes out to be $1060.02864{\mathrm{cm}}^3$.

Calculate the Mass

Now we use the formula Mass = Density * Volume to calculate the mass of the iron block. Substituting the density value and the calculated volume into the formula, we get $Mass=7.86{\mathrm{g}/\mathrm{cm}}^3\times 1060.02864{\mathrm{cm}}^3$.

Compute the Mass

After multiplying these values, the mass comes out to be $8328.30474\mathrm{g}$, or rounded to the nearest hundredth, $8328.30\mathrm{g}$.

Key concepts

Density

Density is a fundamental concept in science used to describe how much mass is contained in a given volume. It is usually expressed in units like grams per cubic centimeter ${\text{(g/cm}}^3\text{)}$ or kilograms per cubic meter ${\text{(kg/m}}^3\text{)}$. Density is defined by the formula:$$\text{Density}=\frac{\text{Mass}}{\text{Volume}}$$This formula tells us that density is simply the mass of the object divided by its volume. In our example, the iron block has a density of $7.86{\text{ g/cm}}^3$. This means that every cubic centimeter of iron weighs 7.86 grams.
Understanding density helps determine whether an object will sink or float in a fluid. Higher density objects tend to sink, while lower density objects can float. Additionally, the concept is critical in applications like material science where knowing the density can help identify substances and design structures.

Volume of a Cuboid

The notion of volume is essential for understanding how much space an object occupies. A cuboid is a 3-dimensional rectangle, akin to a box. To calculate its volume, we multiply its length, breadth (width), and height. The formula is:$$\text{Volume}=\text{length}\times \text{width}\times \text{height}$$For the iron block in this problem, the dimensions are 52.8 cm in length, 6.74 cm in width, and 3.73 cm in height. Substituting these values:$$\text{Volume}=52.8\times 6.74\times 3.73\approx 1060.03{\text{ cm}}^3$$A cuboid's volume tells us the amount of space it occupies in three dimensions. This property is crucial not just in math, but also in fields like engineering and manufacturing where precise volume calculations ensure proper fit and function.

Iron Block

Iron is a commonly used metal with unique properties, making it essential in various industries. One of these properties is its density, mentioned as $7.86{\text{ g/cm}}^3$. This specific density allows us to calculate the mass of any iron object when its volume is known.Iron is used in construction, manufacturing, and many other areas due to its strength and durability. These properties combined with its relatively high density make it suitable for heavy-duty applications. In this exercise, knowing the density of the iron block allows us to tailor calculations in order to assess its mass effectively.
When dealing with iron objects, it's vital to consider the dimensions and how the shape, such as a cuboid, impacts its use and potential. The characteristics of the cuboid shape, combined with the material properties, are instrumental in defining practical applications.

Dimensional Analysis

Dimensional analysis is a mathematical tool that helps ensure the correctness of equations and calculations by focusing on the units of measurement. This approach can identify errors early by comparing the dimensions on both sides of an equation.Consider the exercise: we calculated both the volume in ${\text{cm}}^3$ and the mass using ${\text{g/cm}}^3$, leading us naturally to grams $(\text{g})$ for mass, since:$$\text{Units for mass}=({\text{Density in g/cm}}^3)\times ({\text{Volume in cm}}^3)=\text{g}$$Dimensional analysis was vital in confirming that multiplying density by volume gives us mass. This type of analysis not only aids in problem-solving but is also essential in verifying units in physics, chemistry, and engineering. Knowing how to analyze dimensions allows students and professionals alike to avoid misunderstandings regarding unit conversions and ensures accurate results.