# GCSE( how to find the volume of a pencil)

In the figure below, the dimensions of an unsharpened pencil are given. The unsharpened pencil have graphite in a cylindrical shape, surrounded by wood. The diameter of the graphite is 2mm. The density of graphite is $$640\text{ kg/m}^3$$. The density of wood is $$420\text{ kg/m}^3$$. Find the mass of 500 pencils.

My attempt was the following: I found the volume of the wood: $$3.14\times (0.004)^2\times 0.15= 0.000007536$$ The volume of graphite: $$3.14\times (0.001)^2\times 0.15= 0.000000471$$ I found the mass of 1 pencil: $$0.000007536\times 420 + 0.000000471\times 640= 0.0034\\ 0.0034\times 500= 1.7 \text{ kg}$$

I think I messed up the solution!

All help/solutions appreciated.

Thanks

first of all, I'd like to say it would be much easier for you to calculate with, say, $$4.71×10^7$$ instead of $$0.000000471$$. The mistake you're making here is counting the graphite's volume twice. When calculating the volume of the wooden part, you need to subtract the graphite's volume:

$$7.536×10^{-6}-0.471×10^{-7}=10^{-7}×(75.36-0.471)=74.889×10^{-7}=7.4889×10^{-6} m^3$$

Now, you have the density of both materials. That is to say, you know how many $$kg$$ one $$m^3$$ of wood/graphite weighs. From that info, you can calculate the mass of wood/graphite in one pencil:

$$m_{wood}=7.4889×10^{-8}×420=3145.338×10^{-8}=3.145338×10^{-5}kg$$
$$m_{graphite}=0.471×10^{-7}×640=301.44×10^{-7}=3.0144×10^{-5}kg$$
$$m_{pencil}=m_{wood}+m_{graphite}=10^{-5}×(3.0144+3.145338)=6.159738×10^{-5}kg$$

Now getting the mass of 500 pencils is simple! The answer is

$$6.159738×500×10^{-5}=30.79869×10^{-3}kg \approx 30.8 gr$$

Hopefully this solves your problem :)

You forgot to take out the volume of the graphite when calculating the volume of the wood. In fact the volume of the wood is $$[\pi (0.004)^2(0.15)-\pi (0.001)^2(0.15)]\text{ m}^3$$

• Thank you so much🙏🏻 Jan 9, 2021 at 18:52