Cubical Algebra

Geometry Level 3

You are given two cubes $R$ and $S$ with integer sides of lengths $r$ and $s$ units

If the numerical value of the difference between volumes of the two cubes is equal to the difference in their surface areas, find the value of $\frac {r} {s}$ to 3 decimal places.

The answer is 1.000.

2 solutions

Rohith M.Athreya
Jan 29, 2017

$(r-s)(r^{2}+s^{2}+rs-6(r+s))=0$

$r=s$ and/or

$r(r-6)+s(s-6)=-rs$

either $r$ or $s$ or both are less than 6.

note that only for $r=4$ we get integral solution $s=4$

for $r=1,2,3,5$ $s$ is non integral

so,only possible value $\frac{r}{s}=1$

Saya Suka
Jan 29, 2017

difference in volume = difference in surface area
r^3 - s^3 = 6(r^2 - s^2)
(r - s)(r^2 + rs + s^2) = (r - s) * 6 * (r + s)
r / s = 1,
or
r^2 + rs + s^2 = 6(r + s)
Subtract both sides of the equation by 3rs
(r - s)^2 = 6r + 6s - 3rs = 3[4 - (r - 2)(s - 2)]
LHS >= 0, thus
4 - (r - 2)(s - 2) >= 0
(r - 2)(s - 2) <= 4
Seeing that RHS has a factor of 3, then [4 - (r - 2)(s - 2)] must also have a factor of 3 in it, and if not, then LHS = 0 (leading to an answer of r = s = 4).

The only multiple of 3 which is less than 4 (to keep RHS >= 0) is 3, so (r - 2)(s - 2) = 1 = (+-1) * (+-1)
r - 2 = s - 2 = 1 or r - 2 = s - 2 = -1
r = s = 3 or r = s = 1

So either way, r = s, that gave the answer r/s = 1.000

I'm still don't understand what do you mean by "Seeing that RHS has a factor of 3, then [4 - (r - 2)(s - 2)] must also have a factor of 3 in it, and if not, then LHS = 0"

You found out that $4-(r-2)(s-2) \geq 0$ , then $4-(r-2)(s-2)$ can be $0,1,2,3,4,5,...$ . Why must only factor of $3$ ? It also can be $2$ and $4$ .

Jason Chrysoprase - 4 years, 4 months ago

Because LHS is a square of some integer, and RHS must match the 3 outside the bracket with at least one factor of 3 inside them.

Saya Suka - 4 years, 4 months ago

If r-s=0, then you can't divide r-s from the equations in the beginning.

Vishnu Kadiri - 3 years, 8 months ago

Cubical Algebra

The answer is 1.000.

2 solutions

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