A certain dosage of radiation, measured in kilorads, must be given to the

tumor near the brain. The dose delivered must be sufficient to kill the malignant cells but

the aggregate dose must not exceed established tolerance levels for the brain. Two

beams which would deliver radiation exposure to the cells will be used. The goal is to

select certain beam durations that would generate the best dosage distribution by

minimizing the radiation absorbed by the brain. The data for the radiation therapy is given

below:

Fraction of dose absorbed per second Average dosage Area Beam 1 Brain 0.4 Spine 0.3 Tumor 0.5 Center of Tumor 0.6 Beam 2 [in kilorads) 0.5 0.1 At most 2.? 0.5 At most 6 0,4 At least 6 Determine the optimal exposure times for beam 1 and beam 2