Fold Dv
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Core 1 Maths AS exam paper question help?
3) A diagram shows a rectangular sheet of metal 24 cm by 9cm.
A square of side x cm is cut from each corner and the metal is then folded along the dotted lines to make an open box with a rectangular base and height x cm.
a) Show that the volume, V cm3, of liquid that the box can hold is given by V = 4x^3 - 66x^2 + 216x
b) (i) Find dV/ dx
(ii) Show that any stationary values of V must occur when x2 – 11x + 18 = 0.
(iii) Solve the equation x2 – 11x + 18 = 0.
(iv) Explain why there is only one value of x for which V is stationary.
Firstly, find the length, breadth of the rectangular box.
Length of box = 24 - x - x = 24 -2x
Breadth of box = 9 - x - x = 9 -2x
Depth = x
a) Volume of box, V = length * breadth * height
V = (24 - 2x)(9 - 2x)(x) = (24x - 2x^2)( 9 - 2x)
V = 4x^3 - 66x^2 +216x (shown)
b)(i) dV/dx = 12x^2 - 132x + 216
(ii) To find stationary value of V, dV/dx = 0
12x^2 - 132x + 216 = 0
x^2 – 11x + 18 = 0 (divide by 12) (shown)
(iii) x^2 – 11x + 18 = 0
(x -9 )(x - 2) =0
x = 2 or x =9(rejected)
Therefore, x =2
(iv) The reason why I rejected x=9 is because if x=9, then the breadth will be 9-9-9=-9. Breadth cannot be less than 0. So x has to be 2.
Hope it helps. Cheers 