a. The quantity y is partly constant and partly varies inversely as the square of x. Write down the relationship between x and y
b. The quantity y is partly constant and partly varies inversely as the square of x. When x = 1, y = 11, and when x = 2, y = 5. Find the value of y when x = 4
Verified Correct Answer: .
Step-by-Step Mathematical Solution
a. y = c + k(\(\frac{1}{x^2}\)) where c and k are constant
b. when x = 1, y = 11
∴ 11 = c + k(\(\frac{1}{x^2}\)) = > 11 = c + k --- eqn (I)
when x = 2, y = 5
∴ 5 = c + k(\(\frac{1}{x^2}\)) = > 5 = c + k\(\frac{1}{2^2}\) --- eqn (II)
Multiply through by 4) 20 = 4c + k - - - - - - - -(III)
11 = c + k --- eqn (I)
20 = 4c + k - - - - - - (III)
From eqn(I) k = 11 - c
Put k = 11 - c into eqn(III)
20 = 4c + k = 20 = 4c + 11 - c
20 = 3c + 11
3c = 20 - 11 = 9
3c = 9
c = 3.
Put c = 3 into eqn(I)
11 = c + k
11 = 3 + k
k = 11 - 3 = 8
∴ c = 3, k = 8.
When x = 4,
But c = 3, K = 8
y = 3 + \(\frac{8}{4^2}\)
y = 3 + \(\frac{1}{2}\)
y = \(\frac{7}{2}\).
Source: Myschool Past Questions Archive (Record #75920)