A beam of particles is directed at a 0.017-kg tumor. There are 2.4 x 1010 particles per second reaching the tumor, and the energy of each particle is 5.2 MeV. The RBE for the radiation is 14. Find the biologically equivalent dose given to the tumor in 27 s.

To find the biologically equivalent dose given to the tumor in 27 seconds, we can use the following formula:

Biological Equivalent Dose = Dose × RBE × N,

where
Dose = Energy per particle × Number of particles per second × Time, and
N = Number of seconds.

Let's plug in the values given in the problem:

Energy per particle = 5.2 MeV = 5.2 x 10^6 eV
Number of particles per second = 2.4 x 10^10 particles/second
Time = 27 seconds
RBE = 14

First, let's calculate the dose:

Dose = (5.2 x 10^6 eV/particle) × (2.4 x 10^10 particles/second) × (27 seconds)
= (5.2 x 10^6 eV/particle) × (2.4 x 10^10 particles/second) × (27 seconds)
= (5.2 x 10^6 x 2.4 x 10^10 x 27) eV

Now, let's calculate the biological equivalent dose:

Biological Equivalent Dose = (5.2 x 10^6 x 2.4 x 10^10 x 27) eV × 14
= (5.2 x 2.4 x 27) x (10^6 x 10^10 x 14) eV
= (141.12) x (10^6 x 10^10 x 14) eV
= 141.12 x 10^20 eV

Therefore, the biologically equivalent dose given to the tumor in 27 seconds is 141.12 x 10^20 eV.