Past Papers’ Solutions  Assessment & Qualification Alliance (AQA)  AS & A level  Mathematics 6360  Pure Core 1 (6360MPC1)  Year 2017  June  Q#8
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Question
The water level in a reservoir rises and falls during a fourhour 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 ;
It is a quadratic equation.
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;
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