SAT1000 - P606

Geometry Level pending

As shown above, in $\triangle ABC$ , $AB=BC=2$ , $\angle ABC = \dfrac{2\pi}{3}$ .

If $P$ is outside plane $ABC$ and point $D$ is on segment $AC$ , so that $PD=DA, PB=BA$ , then find the maximum volume of pyramid $PBCD$ .

Let $V$ denote the volume of $PBCD$ , submit $\lfloor 10000V \rfloor$ .

Have a look at my problem set: SAT 1000 problems

The answer is 5000.

1 solution

A Former Brilliant Member
Jun 9, 2020

The volume will be maximum when $D$ is the mid point of $\overline {AC}$ and $P$ is directly above $D$ . Then the volume will be $V=\dfrac {1}{3}\times \dfrac{1}{2}\times \sqrt 3\times 1\times \sqrt 3=0.5$ . ( $|\overline {AC}|=2\sqrt 3$ , height of $\triangle {ABC}$ with base $\overline {AC}$ is $1$ ).

So, $\lfloor 10000V\rfloor =\boxed {5000}$ .

SAT1000 - P606

The answer is 5000.

1 solution

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