Hello ceceeee Originally Posted by

**ceceeee** how do you calculate the depth?

QUESTION:

A water trough has a uniform cross section in the shape of half an ellipse. The width of the trough is 80 cm and the areas of its cross section is 1890cm².

1) The trough is 300cm long. Calculate the number of litres of water the trough will hold when it is full.

2) Calculate the depth of the trough. Give your answer to the nearest centimetre.

Please help

1) The volume of the trough is its cross-sectional area multiplied by its length. So that's:$\displaystyle 1890 \times 300=567000 \text{ cm}^3$$\displaystyle =567$ litres

2) If the semi-axes of an ellipse are of lengths $\displaystyle a$ and $\displaystyle b$, then its area is $\displaystyle \pi ab$. The width of this trough is $\displaystyle 80$ cm; so $\displaystyle a = \tfrac12.80=40$. The area of a whole ellipse would be $\displaystyle 2 \times 1890 = 3780\text{ cm}^2$. So if the depth of the trough (which is half the length of the other axis) is $\displaystyle b$, then:$\displaystyle \pi ab = 3780$

$\displaystyle \Rightarrow b = \frac{3780}{40\pi}$$\displaystyle =30$

So the depth of the trough is $\displaystyle 30$ cm.

Grandad