Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Feb-Mar | (P1-9709/12) | Q#3

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Question

The diagram shows a water container in the form of an inverted pyramid, which is such that when  the height of the water level is h cm the surface of the water is a square of side  cm.

     i.       Express the volume of water in the container in terms of h.

[The volume of a pyramid having a base area A and vertical height h is .]

Water is steadily dripping into the container at a constant rate of 20 cm3 per minute.

   ii.       Find the rate, in cm per minute, at which the water level is rising when the height of the water  level is 10 cm.

Solution


i.
 

It is evident from the diagram that water in the container is in the shape of an inverted pyramid.

Expression for the volume of a pyramid having a base area A and vertical height h is;

Therefore, volume of water will be;

We are given that when the height of the water level is h cm the surface of the water is a square of  side  cm.

   ii.
 

We are required to find the rate, in cm per minute, at which the water level is rising when the height  of the water level is 10 cm.

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

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;

Therefore, we are required to find;

We are given that water is steadily dripping into the container at a constant rate of 20 cm3 per  minute. This translates to change in volume of water at a constant rate of 20 cmper minute.  Therefore;

We know that;

Therefore;

Since we are looking for , we need;

We have from (i) that;

Therefore we can find ;

Rule for differentiation of  is:

Gradient (slope) of the curve at the particular point is the derivative of equation of the curve at that  particular point.

Gradient (slope)  of the curve  at a particular point  can be found by substituting x- coordinates of that point in the expression for gradient of the curve;

Therefore;

We are given that , therefore;

Therefore, at  we have .

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