3.7: Triangular prism inside a similar pyramid

0606/22/F/M/22 — Question 12 · 7 marks

In this question all lengths are in centimetres.

The diagram shows a right triangular prism of height

h

inside a right pyramid of height

12

with equilateral base

ABC

of side

8

. The base of the prism sits on the base of the pyramid. Points

P

Q

and

R

lie on

OA

OB

and

OC

respectively, and pyramids

OABC

and

OPQR

are similar.

(a)

Show that the volume,

V

, of the triangular prism is given by

V = \dfrac{\sqrt{3}}{9}\bigl(ah^{3} + bh^{2} + ch\bigr)

where

a

b

and

c

are integers to be found.

[4]

(b)

It is given that, as

h

varies,

V

has a maximum value. Find the value of

h

that gives this maximum value of

V

[3]