Shrinking Cubes

Algebra Level 1

I have a cuboid with side lengths p p cm, q q cm, r r cm. It has the special property that its surface area (in c m 2 cm^2 )is exactly twice the volume (in c m 3 cm^3 ). I know that if I were to shorten each side length by 1 1 cm the volume would be 100 c m 3 100cm^3 .

Evaluate the sum of the side lengths p + q + r p +q +r .


The answer is 101.

Eamon Gupta by Eamon Gupta , 20, United Kingdom

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1 solution

Eamon Gupta
Jan 2, 2018

We have 2 ( p q + q r + p r ) = 2 p q r 2(pq+qr+pr)=2pqr

p q r − p q − q r − p r = 0 pqr -pq-qr-pr=0

Adding p + q + r − 1 p+q+r-1 to both sides:

( p − 1 ) ( q − 1 ) ( r − 1 ) = p + q + r − 1 \color{#D61F06}(p-1)(q-1)(r-1)\color{#333333}=p+q+r-1

p + q + r = 100 + 1 = 101 p+q+r=\color{#D61F06}100\color{#333333}+1=\boxed{101}

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