The key to related rates is nearly always the chain rule, and this is what you haven't explicitly invoked. You might want to try filling up this pattern...
... where straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case time), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).
So what have we got here? x the opposite side of the right triangle, 0.5 the adjacent, z the hypotenuse, dx/dt the rate of increase of x, and Pythagoras relating x and z as here...
So differentiate with respect to the inner function, and the inner function with respect to t...
and sub in the given values of z and dx/dt.
Similar reasoning for the angle, lets call it theta.
Don't integrate - balloontegrate!
Balloon Calculus: Gallery
Balloon Calculus Drawing with LaTeX and Asymptote!