14 A 3000 cm long water trough has the shape of a triangular prism. Its cross-section is an isosceles
triangle with apex 90°, and the trough is built with the apex at the bottom.
a i Find the area of an isosceles triangle with apex angle 90°, and height h centimetres.
ii Find the volume of water in the water trough when the water height is h cm.
b The water trough was initially empty, and then was filled with water at a constant rate of
27 litres per minute.
i Find the height h cm of the water after t minutes, and the rate the water was rising.
ii Find the water height, and the rate the water was rising, after 25 minutes.
iii Find when the water height was 30 cm, and the rate the water was then rising.
14 a i Area = $h^2 cm^2$
b i $h = 3\sqrt{t}$, $\frac{dh}{dt} = \frac{3}{2\sqrt{t}}$
ii h = 15 cm, $\frac{dh}{dt} = \frac{3}{10}$ cm/s
iii 100 seconds, $\frac{3}{20}$ cm/s
ii Volume = $3000 h^2 cm^3$
