Let the height of the rectangle be h, and the diameter of the circle be x. The base of the rectangle is obviously equal to the diameter of the circle.
The semicircle has a radius (r) of x/2
Circumference of the semicircle = (pi/2)x
Perimeter is: P = (pi/2)x + x + 2h = 30
2h = 30 - ((pi + 2)/2)x
h = 15 - ((pi + 2)/4)x
The area of a semicircle is pi * r^2 / 2, or pi * ((x/2)^2) / 2, or (pi/8)x^2
An area of a rectangle, base * height, is x * h.
The total area is, therefore, (pi/8)x^2 + xh
Since we already solved for h, we can plug this back in to this, giving:
(pi/8)x^2 + x(15 - ((pi + 2)/4)x)
You can simplify this, but it's really not necessary. All you have to do, since you can pretty easily see that this will be a parabola, is graph that function. Yes, it's a bitch to enter on the calculator, but it's doable (there is only 1 variable, so simplifying it further won't do anything anyways, in terms of graphing).
Basically, when you graph this (Gcalc.net), you see this:
If you have a graphing calculator, now would be a good time to use one. If not, well, I'm not going to explain how to find the limit (x) of a quadratic, but you will find that the point where the parabola shifts (the vertex) is (8.4014861, 63.011155). So, the dimension for x = 8.4014861. To make sure we don't make any rounding errors early on, let's leave it as 8.401.
So, now that we have x, we can plug this back in for h;
h = 15 - ((pi + 2)/4)*8.401
h = 15 - (1.28539816*8.401)
h = 4.201370058
So, rounding to the nearest 10th, the width of the rectangle (or diameter of the circle) is going to be 8.4, and the height of the rectangle is going to be 4.2.