A right conical piece of metal has base radius 1.8 centimeters and height 2.7 centimeters and mass 63 grams. Determine the density of the metal to the nearest hundredth.
The density of any object can be calculated by dividing its mass by its volume. You might’ve seen this relationship demonstrated by the triangle shown. In this question, the mass of our metal is measured in grams. As our dimensions are in centimeters, the volume will be measured in cubic centimeters or centimeters cubed. This means that the units for density in this case will be grams per centimeter cubed.
Our metal in this question is in the shape of a cone. It has a base radius of 1.8 centimeters, and it has a height of 2.7 centimeters. The volume of any cone can be calculated using the formula one-third 𝜋𝑟 squared multiplied by ℎ. Substituting in our values gives us one-third multiplied by 𝜋 multiplied by 1.8 squared multiplied by 2.7. As multiplication is commutative, we can do this in any order. One-third of 2.7 is 0.9, and multiplying this by 1.8 squared is 2.916. The volume of the cone is therefore 2.916𝜋.
In order to get as accurate an answer as possible, we will leave the answer in terms of 𝜋 at present. Our units for volume are cubic centimeters. We can then calculate the density of the piece of metal by dividing 63, the mass, by 2.916𝜋. Typing this into the calculator gives us 6.877065 and so on. We’re asked to round our answer to the nearest hundredth. This is the same as rounding to two decimal places. The deciding number is the seven in the thousandths column. If the deciding number is five or greater, we round up.
The density of the conical piece of metal is 6.88 grams per cubic centimeter to the nearest hundredth.