Question

7. It is given that y is inversely proportional to x². When x = a, y=4 and when x = a + 2,
y=1, where a is a positive constant.
(a) Express y in terms of x.
(b) Ifx increases by 38%, find the percentage change in y, correct to 3 significant figures.​

Answer 1

Answer:

(a) y = 16 / x²

(b) y decreases by 47.5%

Step-by-step explanation:

Given:

• y is inversely proportional to x²  ⇒  y = k / x² for some constant k
• y=4 when x=a  ⇒  4 = k / a²  ⇒  k = 4a²
• y=1 when x=a+2  ⇒  1 = k / (a+2)²  ⇒  k = (a+2)²

From the last two lines,

      4a² = (a+2)²

 ⇒ 4a² - (a+2)² = 0

 ⇒ (2a)² - (a+2)² = 0

 ⇒ ( 2a - (a+2) ) ( 2a + (a+2) ) = 0

 ⇒ ( a - 2 ) ( 3a + 2 ) = 0

 ⇒  a = 2  or  a = -2/3

As it is given that a is positive, it follows that a = 2.

Then k = 4a² = 4 × 2² = 4×4 = 16.  So...

Answer to part (a) is  y = 16 / x².

Let the new values of x and y be written as x' and y'.

We are given that x' is 38% more than x  ⇒  x' = 1.38 x

Then

  y' / y  =  ( 16 / (x')² ) / ( 16 / x² )

           =  ( 16 / (x')² ) × ( x² / 16 )

           = x² / (x')²

           = ( x / x' )²

           = ( 1 / 1.38 )²

           ≈ 0.525

           = 1 - 0.475

That is,  y' = y  -  0.475y  =  y  - ( 47.5% of y ).

Answer to part (b) is that y decreases by 47.5%.

Hope that helps!