Question

Y is inversely proportional to the square of x.

A table of values for x and y is shown.
x= 1, 2, 3, 4
y=4, 1, 4/9, 1/4

a) Express y in terms of x.
b) Work out the positive value of x when y = 25

Answer 1

see explanation

Explanation:

(a)

given y is inversely proportional to x² , then the equation relating them is

y =
`(k)/(x^2)` ← k is the constant of proportionality

to find k use the ordered pair x = 1, y = 4 from the table , then

4 =
`(k)/(1^2)` =
`(k)/(1)`, so

k = 4

y =
`(4)/(x^2)` ← equation of proportion

(b)

substitute y = 25 into the equation and solve for x

25 =
`(4)/(x^2)` ( multiply both sides by x² )

25x² = 4 ( divide both sides by 25 )

x² =
`(4)/(25)` ( take square root of both sides )

`√(x^2)` = ±
`\sqrt{(4)/(25) }` , that is

x = ±
`(√(4) )/(√(25) )` = ±
`(2)/(5)`

the positive value of x is then x =
`(2)/(5)`

Answer 2

Explanation:

we shall construct a relation between x and y. let k be the constant of proportionality.

`y=(k)/(x^2)`

now, we use the values from the table. letting
`(x,y)=(1,4)`, we have

`4=(k)/(1^2)\implies k=4`

we can verify that this works for the other values on the table. hence,

`y=(4)/(x^2)` is our relation.

next, we let
`y=25` and now to find x.

`25=(4)/(x^2)\implies x^2=(4)/(25)\implies x=\pm(2)/(5)`

thus, the positive value of x is
`(2)/(5)`.