A piece of wire of length $20$cm is cut into $2$ parts. the first part is bent into a circle of radius $r$ in cm, the second into a square of side length $s$ in cm.
a) write down an expression for the sum of the perimeters of the two shapes in terms of r and s. use this to express $s$ in terms of $r$
I have got $2πr+4s=20$ but don't even know if this is right or not
b) find an expression for $S$, the sum of the areas enclosed by the two shapes in terms of r
c) use differentiation to determine the value of $r$ for which $S$ is a minimum
Really struggling with this as all other examples ask for minimum and maximum areas. I can't even figure out where to start so would appreciate any help!
Your answer to a is the first step. Now you should solve it for one of the two variables. $s$ will work better in what follows. For b, what is the area of a square of side $s$? What is the area of a circle of radius $r$? Add them together to get the total area. Now substitute the expression you got in a for $s$ and you have the total area as a function of $r$. The first equation shows you the relation between $s$ and $r$ to use up all the wire. As you increase $s$, you must decrease $r$.