Question

Write a general formula to describe the variation. M varies directly with the square of d and inversely with the square root of x; M=12 when d=3 and x=4

Answer 1

Given that 'M' varies directly with the square of 'd',

`M\propto d^2`

Given that 'M' varies inversely with the square root of 'x',

`M\propto\frac{1}{\sqrt[]{x}}`

Combining the relationships,

`M\propto\frac{d^2}{\sqrt[]{x}}`

Let 'k' be the constant of proportionality. Then,

`M=k\cdot\frac{d^2}{\sqrt[]{x}}`

Given that M=12 when d=3 and x=4,

`\begin{gathered} 12=k\cdot\frac{(3)^2}{\sqrt[]{4}} \\ 12=k\cdot(9)/(2) \\ k=(12\cdot2)/(9) \\ k=(8)/(3) \end{gathered}`

Substitute the value of this constant in the general expression,

`M=(8)/(3)\cdot\frac{d^2}{\sqrt[]{x}}`

Thus, the required general formula to describe the relation is obtained as,

`M=(8)/(3)\cdot\frac{d^2}{\sqrt[]{x}}`