z varies directly as x^2 and inversely as y^2. If z=106 when x=4 and y=4. find z if x =9 and y=8

Question

z varies directly as x^2 and inversely as y^2. If z=106 when x=4 and y=4. find z if x =9 and y=8 (Round your number to the nearest hundredth)

Answer 1

z varies directly as x**2 --> z = k1*x**2

z varies inversely as y**2 --> z = k2/y**2

if z = 106 when x=4, then 106 = k1 * 4**2, and k1 = 17.67

if z = 106 when y=4, then 106 = k2/4**2, and k2 = 41.18

so the relationships become z = 17.67 * x**2

z = 41.18/y**2

when x = 9

z = 17.67 * 9**2 = 1431.27

when y = 8

z = 41.18/8**2 = 0.643

Answer 2

Since z varies directly as x² and inversely as y² it follows that z must have the form:

z=cx²/y²

where "c" is some constant which we can find by recalling that z=106 when x=y=4

106=c*4²/4²=c -> c=106

Therefore:

z=106x²/y²

To find the value of z when x=9 and y=8 we simply plug in those values of x and y into the equation we have derived for z:

z=106*9²/8²=106*81/64≈134.16