can YOU help with rational and radical functions, please?

Question

Write a specific formula to describe the variation: S varies jointly with the inverse of the cube root of P and inversely with the square root of L; S = 2 when P = 8, L = 25.

Answer 1

Inverse variation :

If x and y are two variables, then the direct variation is y = kx, where, k is some nonzero constant.

If x and y are two variables, then the inverse variation is y = k / x , where, k is some nonzero constant.

The non zero constant k is called the constant of variation, and y is said to vary inversely with x.

Observe the equations for the relationships :

RELATIONSHIP	EQUATION
a. y varies directly with x	y = kx
b. y varies inversely with x	y = k / x
c. z varies jointly with x and y	z = kxy
d. y  varies inversely with the square of x	y = k / x ^2
e. z varies directly with y and inversely with x	z = ky / x

Write the specific formula to describe the variation : S varies jointly with the inverse of the cube root of P and inversely with the square root of L.

S = k / [P ^(1/3) * L ^(1/2)].

Substitute the values of S = 2, P = 8, and L = 25 in S = k / [P ^(1/3) * L ^(1/2)].

2 = k / [8^(1/3) * 25^(1/2)]

2 = k / [2 * 5]

⇒ k = 10 * 2 = 20.

If k = 20, then S = 20 / [P ^(1/3) * L ^(1/2)].

Therefore, the variation S = 20 / [P ^(1/3) * L ^(1/2)].