P2

Geometry Level pending

A pencil is made in the shape of an hexagonal prism with a bottom side length of 3 m m 3 mm and a height of 200 m m 200 mm . The pencil body is made of wood and the center is made of graphite. The center is made in the shape of a cylinder with the same height as the pencil, and its bottom sides are circles with radius 1 m m 1mm . Assume 1 m 3 1 m^{3} of wood costs a a (million $), 1 m 3 1 m^{3} of graphite costs 6 a 6a (million $). Which of the following values is closest to the cost of making a single pencil

8 , 45 a 8,45a 7 , 82 a 7,82a 84 , 5 a 84,5a 78 , 2 a 78,2a

Son Nguyen by Son Nguyen , 20, Vietnam

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1 solution

Volume of graphite used in one pencil = π ( 1 m m ) 2 ( 200 m m ) = 200 π m m 3 \pi (1\,mm)^2(200\,mm) = 200\pi \,mm^3

Volume of one pencil = Area of base × \times height

\hspace{63pt} = 6 3 4 6\Large\frac{\sqrt{3}}{4} ( 3 m m ) 2 ( 200 m m ) = 2700 3 m m 3 (3\,mm)^2(200\,mm) = 2700\sqrt{3}\,mm^3

Volume of wood used in one pencil = Volume of pencil - Volume of graphite used in one pencil

\hspace{135pt} = ( 2700 3 − 200 π ) m m 3 (2700\sqrt{3} - 200\pi)\,mm^3

Cost of graphite used in one pencil = ( 200 π ) ( 6 a ) 1 0 9 \Large\frac{(200\pi)(6a)}{10^9}\, million $

Cost of graphite used in one pencil = ( 2700 3 − 200 π ) ( a ) 1 0 9 \Large\frac{(2700\sqrt{3} - 200\pi)(a)}{10^9}\, million $

So, total cost of one pencil = ( 6 ⋅ 200 π + 2700 3 − 200 π ) a 1 0 9 \Large\frac{(6\cdot 200\pi + 2700\sqrt{3} - 200\pi)a}{10^9} ⋅ 1 0 6 \cdot 10^6 $

≈ 7.82 a \hspace{100pt} \approx 7.82a $

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