# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/12) | Q#3

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**Question**

Fig. 1 shows an open tank in the shape of a triangular prism. The vertical ends ABE and DCF are identical isosceles triangles. Angle ABE = angle BAE = 30^{o}. The length of AD is 40 cm. The tank is fixed in position with the open top ABCD horizontal. Water is poured into the tank at a constant rate of 200 cm^{3} s^{−}^{1}. The depth of water, t seconds after filling starts, is h cm (see Fig. 2).

** i. **Show that, when the depth of water in the tank is h cm, the volume, V cm^{3}, of water in the tank is given by .

** ii. **Find the rate at which h is increasing when h = 5.

**Solution**

i.

It is evident from the diagram that;

Let’s first find area of .

Expression for the area of the triangle is;

Consider the diagram below.

Consider .

Expression for trigonometric ratio in right-triangle is;

Hence;

Finally;

ii.

It is evident that we are required to find .

Rate of change of with respect to is derivative of with respect to ;

Rate of change of with respect to is derivative of with respect to ;

Therefore;

From (i) we have;

Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to is:

Rule for differentiation of is:

Rate of change of with respect to at a particular point can be found by substituting x- coordinates of that point in the expression for rate of change;

Rate of change of volume when ;

We are given that;

Finally;

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