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 an...
Question 1 Report
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.
Answer Details
(a) "Partly constant and partly varies inversely as the square of \(x\)" means \[y = a + \frac{b}{x^2},\] where \(a\) and \(b\) are constants.
(b) Using the given values:
When \(x = 1, y = 11\): \(a + b = 11.\quad(1)\)
When \(x = 2, y = 5\): \(a + \dfrac{b}{4} = 5.\quad(2)\)
Subtract \((2)\) from \((1)\): \(b - \dfrac{b}{4} = 6 \Rightarrow \dfrac{3b}{4} = 6 \Rightarrow b = 8.\) Then \(a = 11 - 8 = 3.\)
So \(y = 3 + \dfrac{8}{x^2}.\) When \(x = 4\): \[y = 3 + \frac{8}{16} = 3 + 0.5 = \mathbf{3.5}.\]
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