Related rates nearly always depend on the chain rule, so 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).

Here, the two variables to relate are the angle A and shadow length x...

So differentiate with respect to the inner function, and the inner function with respect to t, and sub in the given values of x and dA/dt...

Spoiler:

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Don't integrate - balloontegrate!

Balloon Calculus: Gallery

Balloon Calculus Drawing with LaTeX and Asymptote!