Given that a, b and c are positive real numbers that satisfy b = 64...

Logan Wyatt

Logan Wyatt

Answered

2022-07-08

Given that a, b and c are positive real numbers that satisfy
b = 64 a a 2 64 = 81 c 2 c 2 81 = a 2 + c 2 , find b.

Answer & Explanation

vrtuljakwb

vrtuljakwb

Expert

2022-07-09Added 13 answers

Squaring yields two polynomial equations in a and c,
a 6 + a 4 c 2 128 a 4 128 a 2 c 2 + 4096 c 2 = 0 ,
and
81 a 2 c + 128 a c 2 5184 a + 5184 c = 0.
Over the complex numbers all solutions can be computed by using Groebner bases. Among them the positive real solutions are ( a , b , c ) = ( 0 , 0 , 0 ), ( a , b , c ) = ( 24 / 5 , 30 / 5 , 18 / 5 ). All other solutions are either real with one of the values a , b , c negative, or non-real solutions. We have b 2 = a 2 + c 2 , which is, up to a factor here 30 2 = 24 2 + 18 2 , i.e., 5 2 = 4 2 + 3 2 .
Jamison Rios

Jamison Rios

Expert

2022-07-10Added 6 answers

Great expert answer!

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