Final answer:
The general formula for the variation where m varies jointly with the inverse of the square of b and directly with the cube of b can be expressed as m = k · b^{3} · ⅓b^{2}, which simplifies to m = k · b.
Step-by-step explanation:
The student is asking for a general formula that describes a relationship where m varies jointly with the inverse of the square of b and directly with the cube of b.
A joint variation means that m is directly proportional to more than one variable. In this case, m is directly proportional to the cube of b, and inversely proportional to the square of b. Combining these relationships, we get the following formula:
m = k · ⅓b^{3} where ⅓ means the inverse and k is the constant of proportionality.
To clarify, the cube of b is b raised to the power of 3, denoted as b^{3}, and the inverse of the square of b is 1 divided by the square of b, denoted as b^{-2} or ⅓b^{2}. Hence, the complete general formula considering both direct and inverse variations will be:
m = k · b^{3} · ⅓b^{2}
Since ⅓b^{2} is equivalent to b^{-2}, you can also write this as:
m = k · b^{1}