It is worked out that a solid gold crown has a volume of 150 cubic centimeters. Find the mass of the crown using a value of 19300 kilograms per cubic meter for the density of gold. Give your answer to two significant figures.
Okay. So, let’s say we have this pretty-awesome-looking solid gold crown. And we’re told that this crown has a volume, we’ll call it 𝑣, of 150 cubic centimeters. We’re also told what the density of gold is, 19300 kilograms per cubic meter. And based on this information, we wanna solve for the mass of the crown. To get started doing this, we can recall a relationship that connects volume, mass, and density.
That equation tells us that the density of an object, symbolized by the Greek letter 𝜌, is equal to the mass of that object divided by the space it takes up, its volume. This equation shows us that if we want to isolate the mass of the object 𝑚 on one side of this expression, then we can do that by multiplying both sides of the equation by the volume 𝑣 of our object. When we do this, we find the result that the mass of an object is equal to its density multiplied by its volume. Because our crown is made of solid gold, and we’re given the density of gold, we know 𝜌. And we’re also told the volume that the crown takes up.
By the way, if you’re wondering how we would find the volume of such an unusual shape as this crown, a fairly common method for finding the volume of an irregularly shaped object is to put it in a known volume of water in a graduated container. Then once the object is in the water, however much the water goes up is a measure of the change in volume that the object caused. And that change in volume is the volume of the object itself.
In any case, we know both the density of our crown as well as the volume of it. So, we’re ready to substitute in and solve for mass. But when we plug these numbers into the equation, we notice that they’re not quite ready to be multiplied together. The reason for that is their units don’t agree. In our density, we have units of cubic meters for volume. But in our volume, we have units of cubic centimeters. We’ll need to get these two different units for volume on the same footing before we multiply these numbers together.
We could choose to change either one of these units. But just to pick one, let’s change the cubic centimeters to cubic meters. To begin doing that, we recall that one meter is equal to 100 centimeters. And then, if we cube both sides of this equation, multiplying each side by itself twice over, then we find that one meter cubed is equal to 100 cubed times centimeters cubed. If we then go ahead to divide both sides of the equation by 100 cubed, that term cancels on the right-hand side. And we find that one centimeter cubed is equal to one divided by 100 to the third cubic meters.
What we’ve done so far then is we’ve solved for the number of cubic meters that equals one cubic centimeter. But we don’t have one cubic centimeter as the volume of our crown. We have 150. What we can do then is multiply both sides by that amount, 150. And when we do, we see that 150 cubic centimeters is equal to 150 divided by 100 cubed cubic meters.
So, we can now take this value in cubic meters and substitute it in for 150 cubic centimeters. And then with that substitution made, look what happens to the units in this expression. The units of cubic meters in numerator cancel with the units of cubic meters in denominator. In the end, when we calculate all this out, we’ll be left with the units of kilograms, the units of mass.
When we enter this whole expression on our calculator, we get a result of 2.895 kilograms. But this isn’t our final answer. Because we want our answer to two significant figures. So, let’s count off those figures now. Here is one significant figure. Here’s our second. And here’s our third.
To round our answer to two significant figures, we’ll draw a little dividing line between our second and third significant figures. And then, seeing that this third significant figure is a nine, that is, is five or more, we’ll round our second significant figure up one digit. That is, the eight turns into a nine when rounded to two significant figures. And our final answer for the mass is 2.9 kilograms. That’s the mass of this solid gold crown.