# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2017 | June | Q#8

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Question

The water level in a reservoir rises and falls during a four-hour period of heavy rainfall. The height, h cm, of water above its normal level, t hours after it starts to rain, can be modelled by the equation

,

a.   Find the rate of change of the height of water, in cm per hour, 3 hours after it starts to rain.

b.   Find the values of t for which the height of the water is decreasing.

Solution

a.

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;

We are given;

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

Rule for differentiation is of  is:

Rule for differentiation is of  is:

Rule for differentiation is of  is:

Therefore rate of change of h at t=3;

b.

To test whether a function  is increasing or decreasing at a particular point , we  take derivative of a function at that point.

If  , the function  is increasing.

If  , the function  is decreasing.

If  , the test is inconclusive.

From (a) we have found that;

For decreasing height of water ;

We solve the following equation to find critical values of

For a quadratic equation , the expression for solution is;

Now we have two options.

Hence the critical points on the curve for the given condition are  & .

Standard form of quadratic equation is;

The graph of quadratic equation is a parabola. If  (‘a’ is positive) then parabola opens upwards  and its vertex is the minimum point on the graph.
If
(‘a’ is negative) then parabola opens downwards and its vertex is the maximum point on the  graph.

We recognize that equation  is a quadratic equation representing upwards  opening parabola.

Therefore conditions for  are;