Let a, b, c be positive real numbers such that bc = a/2, ca = 8b, and ab = 128c. Find a + b + c.
+499 If you multiply all three of our given expressions together, you get:
\((abc)^2 = (1/2 * 8 * 128)abc\)
Dividing by abc on both sides, we then get:
\(abc = 64 * 8 = 512\)
If we look back to our first expression \(bc = a/2\), we can multiply a on both sides to get a known value(abc).
This gives us:
\(abc = a^2/2\)
\(512 = a^2/2\)
\(a^2=1024\)
Realize that 1024 = 210, so the square root of it is simply 25 = 32
\(a = 2^5 = 32\)
Repeat the same steps for our second equation by multiplying by b on both sides, and we get:
\(abc = 8b^2\)
\(512 = 8b^2\)
\(64 = b^2\)
\(b = 8\)
\(abc = 512\)
Since a = 32 and b = 8
That only leaves us with c, which is then:
\(512/(32*8) = 2\)
a = 32
b = 8
c = 2
a+b+c = 32 + 8 + 2 = 42
+499 If you multiply all three of our given expressions together, you get:
\((abc)^2 = (1/2 * 8 * 128)abc\)
Dividing by abc on both sides, we then get:
\(abc = 64 * 8 = 512\)
If we look back to our first expression \(bc = a/2\), we can multiply a on both sides to get a known value(abc).
This gives us:
\(abc = a^2/2\)
\(512 = a^2/2\)
\(a^2=1024\)
Realize that 1024 = 210, so the square root of it is simply 25 = 32
\(a = 2^5 = 32\)
Repeat the same steps for our second equation by multiplying by b on both sides, and we get:
\(abc = 8b^2\)
\(512 = 8b^2\)
\(64 = b^2\)
\(b = 8\)
\(abc = 512\)
Since a = 32 and b = 8
That only leaves us with c, which is then:
\(512/(32*8) = 2\)
a = 32
b = 8
c = 2
a+b+c = 32 + 8 + 2 = 42
