Power of Maths: Paper 1 – Section 8 (a) Geometry EXAMPLE 23
A rectangular sheet of metal 80 cm × 50 cm is used to form an open box (no lid) by removing squares of side x cm from each corner, as shown, and folding up the fl aps along the dotted lines.
Show that the volume is given by
V = 4 x 3 − 260 x 2 + 4000x. Find the maximum volume of the box.
x
x x
x 80 cm
Solution 1. V (volume) 2. Diagram:
x
xx x
(50 − 2 x) (80 − 2 x) EXAMPLE 24
A farmer has 800 m of fencing and wishes to make an enclosure consisting of four equal area rectangles, as shown.
xx x y y y xx x y x
(a) Express y in terms of x. (b) Express the total area A in terms of x. (c) Find x, if the total area A is to be a maximum.
Solution 1. A (area) 2. Diagram: xx x
y y y xx x y x x y
3. A = 4xy 4. 800 = 8x + 5y 5y = 800 − 8x y = 160 − 8
5. A = 4x(160 − 8 _
A = 640x − 32 __
6. dA ___
640 − 64 __
dx = 640 − 64 __
x = 50 m
There is only one stationary point. This is a local maximum.
(a) y = 160 − 8 _
x y (c) x = 50 m 5 x
(b) A = 640x − 32 __
5 x 2 x x x x 50 cm
3–5. V = (80 − 2x)(50 − 2x)(x) x = (4000 − 260x + 4 x 2 )(x) V = 4 x 3 − 260 x 2 + 4000x
