Question

Able to help Q39? Thank you

Answer 1

A)

y  varies inversely as (x-1).

Implies    y  α  1/(x -1)

                 y =  k/(x-1).        Where k is constant of proportionality.

                 y*(x - 1) = k

when y = 12, x = 1/2

12*(1/2 - 1) = k

12*(0.5 - 1) = k

k = 12*(-0.5) = -6.

k = -6.

Equation connecting x and y

  y*(x - 1) = k,          recall k = -6

  y*(x - 1) = -6

b)

when x = a.

y*(x - 1) = -6

y*(a - 1) = -6

y = -6 / (a -1)

when x = 2a.

y*(2a - 1) = -6

y = -6 / (2a -1)

Difference in y is 1.8

-6 / (2a -1)  -  -6/(a - 1) = 1.8

-6 / (2a -1)  + 6/(a - 1) = 1.8

6 / (a -1) - 6 /(2a -1) = 1.8

6( 1/(a-1) -  1/(2a - 1)) = 1.8

1/(a-1) -  1/(2a - 1) = 1.8/6

1/(a-1) -  1/(2a - 1) = 0.3

((2a - 1) - (a - 1)) / ((a-1)(2a -1)) = 0.3

(2a - 1 -a + 1) /((a-1)(2a -1)) = 0.3

a / (2a² - 3a + 1) = 0.3

a/0.3 = 2a² - 3a + 1

10a/3 = 2a² - 3a + 1

2a² - 3a + 1 = 10a/3

2a² - 3a -10a/3 + 1 = 0

2a² - 19a/3 + 1 = 0

6a² - 19a + 3 = 0

This is a quadratic expression which can be factored.

6a² - 18a - a + 3 = 0

6a(a - 3) - 1(a - 3) = 0

(6a - 1)(a - 3) = 0         

6a - 1 = 0      a = 1/6

a - 3 = 0   a = 3.

a = 3  or  1/6

Since a is positive integer, a = 3 only.