Question

If x varies directly as y and z and inversely as the square

of r, and x = 16 when y = 3, z = 12 and r = 3, find
x when y = 5, z = 30, and r
r =10

Answer 1

x = 6

Explanation:

Take the statement x varies directly as y and z

This means value of x is directly proportional to the value of y and z represented as x ∝ yz which can be represented as x = k.y.z where k is a constant known as the constant of proportionality

If x varies inversely as the square of r we represent this as x ∝ 1/r² or in equation form x = k/r²

Since the value of x is dependent on the values of y, z and jointly we have a single equation of proportionality:

`\bold{x= k\cdot(yz)/(r^(2))}`

Given the values for x, y, z and r we can compute the proportionality constant and determine the value of x for any value of y, z or r

x = 16 when y = 3, z = 12 and r = 3

`16 = k\cdot (3\cdot 12)/(3^2) \\\\16 = k\cdot (3\cdot 12)/(9)\\\\16 = k\cdot4\\\\k = 16/4 = 4\\`

So the constant of proportionality is k = 4. Us this value in the proportionality equation along with y, z and r to get the value of x

`x = 4\cdot (5\cdot 30)/(10 \cdot 10)\\\\x = 4\cdot(150)/(100)\\\\x = (600)/(100) = 6\\\\`

So x = 6 is the answer