Urgent AS Maths help please !

I really need some help on this question, i would really appreciate if someone could guide me through the steps of what to do. Thanks in advance.

Question

A sealed metal container for food is a cyclinder of height 12cm and base radius r cm. Given the surface area of the container must be at most 128pi cm^2.

a) show that r^2 + 12r - 64 </= 0

b) hence find the maximum value of r.

I really need some help on this question, i would really appreciate if someone could guide me through the steps of what to do. Thanks in advance.

Question

A sealed metal container for food is a cyclinder of height 12cm and base radius r cm. Given the surface area of the container must be at most 128pi cm^2.

a) show that r^2 + 12r - 64 </= 0

b) hence find the maximum value of r.

a) Do you know the formula for surface area of a cylinder? The circle ends should be apparent; as for the rest, you essentially have a lot of circumferences making up a certain height,

b) Think about the graph of r^2 + 12r - 64. It will have two roots, inbetween which it is less than zero.

Yh so the formula for surface area of a cylinder is
2Pi r h + 2Pi x r^2

So

2pi x r x 12 + 2pi x r^2 = 128pi ??

I can tell this is wrong but...

I'm not sure what you're doing here. Just divide throughout by 2pi in the equation you had at first and rearrange and the answer follows.

Part (a) should be pretty simple. Just use the equation for the surface area of a cylinder. Look it up if you have to, but it should be quite easy to work out - the top and bottom are just circles, and I'm sure that you know the equation for the area of a circle, and the vertical outside can be cut open to reveal a rectangle of height 12 and base... well, you tell me!

Part (b) is a quadratic inequality, and I suspect it is this that you are really being tested on. I'd go through this process;

- replace the inequality with an equality, so that you have a quadratic equation.
- solve this equation to get two solutions (probably)
- sketch the quadratic, using your two solutions as the intersections with the horizontal axis
- now look at the original inequality and see if the answer suggests itself.

a) Do you know the formula for surface area of a cylinder? The circle ends should be apparent; as for the rest, you essentially have a lot of circumferences making up a certain height,

b) Think about the graph of r^2 + 12r - 64. It will have two roots, inbetween which it is less than zero.

So
r = -16 & r = 4

So would the answer for part b) be 4 ??