YR2 Integral Differentiation Topic Assessment Qn

QNn 6 have no clue what to do:
A log of wood is modelled as a cylinder with radius 10 cm and height 25 cm. It is to be
made into a cuboid with dimensions 2x cm by 2y cm by 25 cm by trimming the cylinder.
The cross-section is shown in the diagram.
(a) Find the value of x for which the area of the rectagle is maximum. [8]
(b) Calculate the volume of the largest cuboid that can be cut from the log. [2]

Saw you are meant to either sub 4xy into og equation of circle (x^2+y^2=100) but have no clue why you do that.
If anyone could give me some pointers as to what to do, and why you do that then that would be very helpful, thanks

Reply 1

mqb2766

A sketch of the circle and the rectangle of 2x by 2y (cuboid cross section) would help. Hopefully youd see you can just consider a point (x,y) in quadrant 1 (positive x and y) which lies on the circle as that corrsponds to an area xy which gives the overall rectangle area. Then its a case of maximising the area such that x,y lie on the circle (so replalce y in the area expression with its y=f(x) from the circle equ/ation) and there are a few ways of proceeding (a couple a bit more elegant/non calculus than the standard way) but differentiating and setting equal to zero isn't that bad.