Taller Math

1. A farmer wishes to enclose a rectangular field using an existing fence for one of the four sides.

(a) Write an expression in terms of x andy that shows the total length of the new fence.
(1)(b) The farmer has enough materials for 2500 metres of new fence. Show that
y = 2500 – 2x
(1)
(c) A(x) represents the area of the field in terms of x.
(i) Show that
A(x) = 2500x – 2x2
(2)(ii) Find A'(x).
(1)
(iii) Hence or otherwise find the value of x that produces the maximum area of the field.
(3)
(iv) Find the maximum area of the field.
(3)
(Total 11 marks)
2. The function gis defined as follows
g: x px2 + qx + c, p, q, c Î
(a) Find g¢(x)
(1)

(b) If g¢(x) = 2x + 6, find the values of p and q.
(2)

(c) g(x) has a minimum value of –12 at the point A. Find(i) the x-coordinate of A;
(ii) the value of c.
(4)
(Total 7 marks)
3. A rectangular piece of card measures 24 cm by 9 cm. Equal squares of length x cm are cut from each corner of the card asshown in the diagram below. What is left is then folded to make an open box, of length l cm and width w cm.

(a) Write expressions, in terms of x, for
(i) the length, l;
(ii) the width, w.
(2)
(b)Show that the volume (B m3) of the box is given by B = 4x3 – 66x2 + 216x.
(1)
(c) Find .
(1)
(d) (i) Find the value of x which gives the maximum volume of the box.
(ii) Calculate the maximumvolume of the box.
(4)
(Total 8 marks)
4. The cost of producing a mathematics textbook is $15 (US dollars) and it is then sold for $x.
(a) Find an expression for the profit made on each book sold.(1)
A total of (100 000 – 4000x) books is sold.
(b) Show that the profit made on all the books sold is
P = 160 000x – 4000x2 – 1500 000.
(3)
(c) (i) Find .
(2)
(ii) Hence calculate the value ofx to make a maximum profit
(2)

(d) Calculate the number of books sold to make this maximum profit.
(2)
(Total 10 marks)
5. A closed box has a square base of side x and height h.
(a) Write...