A tent-shaped shelter has a ridged roof of length 3 metres and a horizontal rectangular floor of length x metres. Each end of the shelter is an isosceles triangle of slant side length 2 metres and base length 1 metre. The shelter, shown in the diagram below, is symmetric about vertical planes through the centre of the rectangular floor and parallel to its sides. The value of x is between 0 and 3
+ sqrt 15. (The shelter cannot exist for other values of x.)
The total volume
V (x)m3 of the shelter is given by
(x)= 1 /12 (2x +3)
−(x −3)2 (0 ≤x ≤3+ 15).
not asked to derive this formula.)
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.
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
x := x between the two parts in your worksheet.
(a) Use Mathcad to obtain the graph of the function
(b) This part of the question requires the use of Mathcad in each sub-part.
(i) By using the differentiation facility and the symbolic keyword
‘simplify’, find an expression for the derivative
(ii) By either applying a solve block or solving symbolically, find a
x for which V (x)= 0.
(iii) Verify, by the Second Derivative Test, that this value of
corresponds to a local maximum of
V (x). (It should be apparent from the graph obtained in part (a) that this is also an overall
maximum within the domain of
(c) Using Mathcad, or otherwise, find the maximum possible volume of
the shelter, according to the model.