A rectangular sheet in a stationery order has area 12 square centimetres. One side measures \(\sqrt{3}\) centimetres. Find the length of the other side. Wri...
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A rectangular sheet in a stationery order has area 12 square centimetres. One side measures \(\sqrt{3}\) centimetres.
Find the length of the other side. Write your answer in the form \(a\sqrt{3}\), where \(a\) is an integer. (3)
Answer Details
The area of a rectangle is length \(\times\) width, so the unknown side is the area divided by the known side:
\[\text{other side} = \frac{12}{\sqrt{3}}\][1]
A surd is not normally left in a denominator, so the fraction is rationalised by multiplying top and bottom by \(\sqrt{3}\). This is multiplying by \(1\), so the value is unchanged, but the bottom becomes a whole number since \(\sqrt{3} \times \sqrt{3} = 3\):
\[\frac{12}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{12\sqrt{3}}{3}\][1]
\[= 4\sqrt{3} \text{ centimetres}\][1]
So \(a = 4\).
Checking by multiplying the two sides back together: \(\sqrt{3} \times 4\sqrt{3} = 4 \times 3 = 12\) cm\(^2\), the given area.
Note that the \(3\) on the denominator divides into the \(12\), not into the \(\sqrt{3}\); the surd is part of the numerator and stays intact. Numerically \(4\sqrt{3} \approx 6.93\) cm, comfortably longer than the \(\sqrt{3} \approx 1.73\) cm side, which is what a \(12\) cm\(^2\) area requires.
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