Answer
It takes $19.4~seconds$ for the detector to measure 420 kJ of energy.
Work Step by Step
We can find the power absorbed by the detector:
$P = (I~cos~\theta)~A$
$P = (1000~W/m^2)~(cos~30.0^{\circ})~(5.00~m)^2$
$P = 21,650.6~W$
We can find the time it takes to measure 420 kJ of energy:
$t = \frac{Energy}{Power} = \frac{420,000~J}{21,650.6~W} = 19.4~s$
It takes $19.4~seconds$ for the detector to measure 420 kJ of energy.
