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Maximum Area of a Quadrilateral

In the quadrilateral PQRS shown below, the side PQ has length 5 metres, the side QR has length 6 metres, and the side RS has length 7 metres. The angle at P is a right angle, and no angle of the quadrilateral exceeds 180◦. The side PS has length x metres, where the value of x is between 0 and 12. (The quadrilateral described cannot exist for other values of x.)

See attached image.

For parts (a) and (b) (and for part (c), if you use Mathcad there) you should provide a printout annotated with enough explanation to make it clear what you have done.
If you define x to be a range variable in part (a) and wish to use x in a symbolic calculation in part (b), then you will need to insert the definition x := x between the two parts in your worksheet. (For more details, see the bottom of page 49 in A Guide to Mathcad.)
(a) Use Mathcad to obtain the graph of the function A(x).
(b) This part of the question requires the use of Mathcad in each sub-part.
(i) By using the differentiation facility, and if you wish the symbolic keyword 'simplify', find an expression for the derivative A′(x).
(ii) By applying a numerical solve block (rather than attempting to solve symbolically), find a value of x for which A′(x) = 0.
(iii) Verify, by the Second Derivative Test, that this value of x corresponds to a local maximum of A(x). (It should be apparent
from the graph obtained in part (a) that this is also an overall maximum within the domain of A(x).)
(c) Using Mathcad, or otherwise, calculate the maximum possible area of the quadrilateral.


Solution Preview

Define equation for A as shown here. The square root is found in your operators or calculator toolbar depending on which version of Mathcad you have. .

Let's pick a range of x values to calculate. It is given that x is between 0 and 12, so we can choose from 0.1 to 11.9 going up by 0.1. This is written as follows (input ; after the 0.2):

Next we insert Plot, x-y Plot. In the y axis we put A(x). In the x axis we put x. The plot will look similar to the one shown here:

Next we want to take the derivative. Let's call this D(x). So we set up the equation shown here. The derivative is in your ...

Solution Summary

The maximum area of a quadrilateral is determined.