The quantity y is partly constant and partly varies inversely as the square of x. (a) Write down the relationship between x and y. (b) When x = 1, y = 11 and when x = 2, y = 5, find the value of y when x = 4.
Explanation
(a) \(y = k + \frac{c}{x^{2}}\) where c and k are constants. (b) When x = 1, y = 11 \(11 = k + \frac{c}{1^{2}} \implies 11 = k + c ... (1)\) When x = 2, y = 5 \(5 = k + \frac{c}{2^{2}} \implies 5 = k + \frac{c}{4}\) \(\equiv 20 = 4k + c ... (2)\) (2) - (1) : \(4k - k = 20 - 11 \implies 3k = 9\) \(k = 3\) Put k = 3 in (1), we have \(11 = 3 + c \implies c = 11 - 3 = 8\) \(\therefore y = 3 + \frac{8}{x^{2}}\) When x = 4, \(y = 3 + \frac{8}{4^{2}} = 3 + \frac{1}{2}\) = \(3\frac{1}{2}\)