And one more question please. This is from the (ATAR Notes topic tests HSC mathematics Edition 1 2017-2019. Page 35 Q9.) I think they accidentally forgot to put the answer for this one. The question is :

A strip of wire 96 cm in length is used to build a square prism. Supposing that the length of the square side is x cm, show that the surface area of the square prism is given by S= 6x(16 - x). Hence, find the volume of the prism with the maximum surface area.

Thanks guys.

Sorry for missing this one!

The question's wording is actually quite ambiguous! However, if you initially thought that the object was a cube, and the surface area should've been 6x

^{2} (like I did!), you would've noticed there would have been no maximum. Hence, the object is just a square-sided rectangular prism with at least one set of opposite sides being squares.

Let the square sides have length x, and the other edge have length y.

Since the total length of wire is 96cm, we have that 8x+4y=96 ie. y=24-2x.

Now, we have the surface area being 2(x

^{2}+2xy).

ie. S=2x(x+2y)

S=2x(x+2(24-2x))

S=2x(48-3x)

S=6x(16-x)

From here, we find that \(\frac{dS}{dx} = 96-12x\) ie. there's a maximum when x=8 (you can test for a maximum in whichever way you like) (and therefore also when y=8! basically the object was a cube the whole time, which makes sense.) And thus the volume which maximises the surface area is just 8x8x8, or 512cm

^{3}.

Hope this helps