2.9: Volume and related rates in a triangular prism

0606/21/M/J/18 — Question 12 · 9 marks

2.9 diagram
In this question all lengths are in metres.
A water container is in the shape of a triangular prism. The cross-section of the water in the container is an isosceles triangle ABCABC, with ∠ABC=∠BAC=30°\angle ABC = \angle BAC = 30\degree. The length of ABAB is xx and the depth of water is hh. The length of the container is 55.
(i)
Show that x=23 hx = 2\sqrt{3}\,h and hence find the volume of water in the container in terms of hh.
[3]
(ii)(a)
The container is filled at a rate of 0.5 m30.5\text{ m}^{3} per minute. At the instant when hh is 0.25 m0.25\text{ m}, find the rate at which hh is increasing.
[4]
(ii)(b)
Find the rate at which xx is increasing.
[2]