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

Answer:

x = 6

Step-by-step 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:

$\bold{x= k\cdot\frac{yz}{r^{2}}}$

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

$16 = k\cdot \frac{3\cdot 12}{3^2} \\\\16 = k\cdot \frac{3\cdot 12}{9}\\\\16 = k\cdot4\\\\k = 16/4 = 4$

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

$x = 4\cdot \frac{5\cdot 30}{10 \cdot 10}\\\\x = 4\cdot\frac{150}{100}\\\\x = \frac{600}{100} = 6$

So x = 6 is the answer