We label the points on the window for convenience:
Perimeter p is given by:
p = circumference of the semicircle + length AB + length BC + length CD
The circumference of a semi-circle is half the circumference of a circle:
`"curve"\ AD = 1/2(2pir) = pi r`
From the question, we know
AB = BC + 30
Also, we can see from the diagram
AD = BC = 2r
So, putting it all together, we have:
p = curve AD + AB + BC + CD
p = (πr) + (2r + 30) + 2r + (2r + 30)
Simplifying, we can write our function as
p = p(r) = π r + 6r + 60
Note the radius cannot be negative, and the window would not exist if `r = 0`.
Also, when we write "p = p(r)", we mean "p, which is a function of r".