A brick has a mass of 3.5 kilograms. It is a rectangular prism with side lengths of 23 centimeters, 11 centimeters, and seven centimeters. What is the density of the brick? Give your answer to the nearest kilogram per cubic meter.
Okay, let’s say this is the brick we’re being asked about in the question. We’re told the mass of the brick as well as the length of the three sides of the brick. These lengths are the length, width, and height of the brick, which we can label with the letters 𝑙, 𝑤, and ℎ, respectively.
It doesn’t matter which of these numbers we call the length, which one we call the width, or which one we call the height. For this question, let’s say that the length is the longest one, so 𝑙 equals 23 centimeters. Let’s say that the width is the middle one, so 𝑤 equals 11 centimeters. And the height is the shortest one, so ℎ equals seven centimeters. Now that we know which length is which, lets add them to our diagram for safekeeping.
Before we go any further, let’s notice that each of these lengths is in centimeters. But the question asked us to give our final answer in kilograms per cubic meter. So before we start calculating anything, let’s convert all of these lengths from centimeters into meters. Since there are 100 centimeters in one meter, this is done by dividing each of our lengths in centimeters by 100. For example, the length of the brick is 𝑙 equals 23 centimeters. And if we divide this by 100, we find the length in meters, which is 0.23 meters. We can do the same for the width and height of the brick. And we find that 𝑤 is equal to 0.11 meters and ℎ is equal to 0.07 meters. Now that we know these lengths in meters, let’s update the labels on our diagram.
The other thing we’re told in the question is the mass of the brick, which is 3.5 kilograms. So if we call this mass 𝑀, then 𝑀 is equal to 3.5 kilograms.
Given all of this information, we’re asked to find the density of the brick. We can do this by recalling that the general formula for the density of an object, which we denote with the Greek letter 𝜌, is that 𝜌 is equal to 𝑀 divided by 𝑉, where 𝑀 is the mass of the object and 𝑉 is the volume of the object.
Now for our brick, we know that the mass is 3.5 kilograms, but we don’t yet know the volume of our brick. So in order to calculate the density of the brick, we first need to find its volume. We’re told that the brick is a rectangular prism, so the volume of the brick can be found with the formula 𝑉 is equal to the length of the brick 𝑙 times the width of the brick 𝑤 times the height of the brick ℎ.
Since we know each of these lengths, we can now simply substitute them into this formula. So the volume of the brick is equal to the length, which is 0.23 meters, times the width, which is 0.11 meters, times the height, which is 0.07 meters. Let’s first multiply the numbers together, so 0.23 times 0.11 times 0.07, which we find to be 0.001771. And then let’s simplify the units, which are meters times meters times meters. So the units here are meters cubed. This means we have found the volume of the brick to be 𝑉 equals 0.001771 meters cubed. And we’re ready to calculate the density of the brick using the general formula for density.
So for the brick, the density 𝜌 is equal to the mass, which we know is 3.5 kilograms, divided by the volume that we have just found. We can simplify this fraction by separating the numerical part from the units of the fraction. So 𝜌 is equal to 3.5 divided by 0.001771 with units of kilograms over meters cubed or kilograms per cubic meter. We can then simplify this numerical part using a calculator. And we find that this fraction is equal to 1976.28, where we are just giving the first two decimal places.
In fact, the question asked us to give our answer to the nearest kilogram per cubic meter. So we can round our answer to the nearest whole number. Doing this rounding gives us our final answer, which is that the density 𝜌 is equal to 1976 kilograms per cubic meter.