A weird tetrahedron? No?

Geometry Level 4

Let $V$ be the volume of a tetrahedron bounded by the coordinate planes and the plane $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ where $a$ , $b$ , $c$ are positive real numbers.

If $V$ can be written in the form of
$\large \frac{2^{\zeta}a^{\alpha}b^{\beta}c^{\gamma} } {144^{\xi}}$

where $\zeta, \alpha, \gamma$ and $\xi$ are positive integers, what is the value of $50\lfloor \zeta + \alpha + \beta + \gamma + \xi \rfloor$ ?

Image Credit: Wikimedia Symmetries of the tetrahedron .

The answer is 200.

1 solution

คลุง แจ็ค
Jul 5, 2015

From the equation given we can know the intersection between the plane and the coordinate.

After visualizing we can treat this tetrahedron like a simple pyramid of height $a$ and base area $\displaystyle \frac{1}{2}bc$

then the volume is $\displaystyle \frac{abc}{6}=\frac{2abc}{12}$

So,

$\displaystyle a=1 , b=1 , c=1,\zeta=1 , \xi =1/2$

$\displaystyle 50\left \lfloor \zeta+a+b+c+\xi \right \rfloor =\boxed{200}$

Great approach! Well done! :)

Noor Muhammad Malik - 5 years, 11 months ago

Shouldn't the answer be 225 then?

Karan Jain - 5 years, 11 months ago

No, you have the solution above that's accurate. Follow it through.

Noor Muhammad Malik - 5 years, 11 months ago

A weird tetrahedron? No?

Image Credit: Wikimedia Symmetries of the tetrahedron .

The answer is 200.

1 solution

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