A rectangular water tank has a base measuring 10 m by 5 m. An engineer needs the length of the diagonal brace across the base. Show that the diagonal, in me...
Question 1 Report
A rectangular water tank has a base measuring 10 m by 5 m. An engineer needs the length of the diagonal brace across the base.
- Show that the diagonal, in metres, is \(5\sqrt{5}\). (3)
- Rationalise the denominator of \(\dfrac{20}{\sqrt{5}}\), giving your answer in simplified surd form. (2)
Answer Details
This question tests using Pythagoras' theorem to express a diagonal length as a surd, and rationalising the denominator of a fraction containing a surd.
(a) The diagonal squared equals the sum of the squares of the two sides of the rectangular base:
\[\text{diagonal}^2=10^2+5^2=100+25=125\]Since \(125=25\times 5\) and 25 is a perfect square:
\[\text{diagonal}=\sqrt{125}=\sqrt{25\times 5}=5\sqrt{5} \text{ m}\ \text{<b>[3]</b>}\](b) Multiply both the numerator and the denominator by \(\sqrt{5}\), so that the denominator becomes a whole number:
\[\frac{20}{\sqrt{5}}=\frac{20\sqrt{5}}{\sqrt{5}\times\sqrt{5}}=\frac{20\sqrt{5}}{5}=4\sqrt{5}\ \text{<b>[2]</b>}\]Rationalising a denominator does not change the value of the fraction, since multiplying top and bottom by the same surd is equivalent to multiplying by 1; it simply rewrites the fraction so that the surd only appears above the line, which is the conventional simplified form.
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