The following is a response to a question that I previously posted, and I unders

Question

The following is a response to a question that I previously posted, and I understand everything except for the part where they solved for 'a' (BOLDED below).

I ONLY need help understanding the steps in solving for 'a' , I understand everything else!

AC = sqrt(1.552 + (3.50 - a)2)

CE = sqrt (a2 + 0.452)

nw = index of refraction of water = 1.33

na = index of refraction of air = 1

using snell's law at C

nw Sini = na Sinr

1.33 a / sqrt(a2 + 0.452) = (3.50 - a) / sqrt(1.552 + (3.50 - a)2)

a = 0.41 (PLEASE SHOW STEPS FOR SOLVING FOR a)

Sini = a / sqrt(a2 + 0.452) = 0.41 / sqrt((0.41)2 + 0.452)

i = Sin-1(0.41 / sqrt((0.41)2 + 0.452) ) = 41.3

Explanation / Answer

1.33 a / sqrt(a2 + 0.452) = (3.50 - a) / sqrt(1.552 + (3.50 - a)2)

1.77a2[1.552 + (3.50 - a)2] = (3.50 - a)2(a2 + 0.452)

2.75a2+1.77a2(3.50 - a)2 = (3.50 - a)2(a2 + 0.452)

2.75a2+(3.50 - a)2 [0.77a2 - 0.452] =0

2.75a2+(12.25-7a+a2)[0.77a2 - 0.452]=0

2.75a2+9.43a2 -5.39a3+ 0.77a4-5.54+3.16a-0.452a2 =0

0.77a4-5.39a3+11.73a2+3.16a-5.54 =0

0.77 *(a +0.699) *(a -0.64) *(a^2 -7.059*a +16.097) =0

a = 0.64

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