A curve has equation \(y=\frac{100}{x^2}\). Calculate the value of \(y\) when \(x=4.5\). Give your answer correct to 3 significant figures.
Question 1 Report
A curve has equation \(y=\frac{100}{x^2}\).
Calculate the value of \(y\) when \(x=4.5\). Give your answer correct to 3 significant figures.
Answer Details
Here \(y\) is inversely proportional to the square of \(x\), so the denominator must be squared before the division is carried out. Getting that order right is the whole skill being tested.
\[ y = \frac{100}{x^2} = \frac{100}{4.5^2} \] [M1] \[ 4.5^2 = 20.25 \] \[ y = \frac{100}{20.25} = 4.9382\ldots \]Correct to \(3\) significant figures, \(y = 4.94\). [A1]
The first three significant figures are \(4\), \(9\), \(3\) and the next digit is \(8\), so the \(3\) rounds up to \(4\).
The frequent calculator error is entering \(100 \div 4.5^{\,}2\) as \((100 \div 4.5) \times 2\) or as \((100 \div 4.5)^2\). Use brackets, \(100 \div (4.5^2)\), so the squaring is confined to the denominator.
Because \(y\) depends on \(x^2\), the curve falls away steeply near the \(y\)-axis and flattens for large \(x\). Tripling \(x\) divides \(y\) by \(9\), which is a quick way to sanity-check any answer from this equation.
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