# Surds Word Problem

## 2012 May June Paper 21/23 Q2

A cuboid has a square base of side $$(2+\sqrt{3}) cm$$ and a volume of $$(16+9 \sqrt{3}) cm ^{3}$$. Without using a calculator, find the height of the cuboid in the form $$(a+b \sqrt{3}) cm$$, where $$a$$ and $$b$$ are integers.

Height $$=h$$
Volume of cuboid

$$V=(2+\sqrt{3})(2+\sqrt{3}) \times h=16+9 \sqrt{3}$$
$$h=\frac{16+9 \sqrt{3}}{4+4 \sqrt{3}+3}$$
$$=\frac{16+9 \sqrt{3}}{7+4 \sqrt{3}} \times \frac{7-4 \sqrt{3}}{7-4 \sqrt{3}}$$
$$=\frac{112+63 \sqrt{3}-64 \sqrt{3}-108}{49-\left(4\sqrt{3})^{2}\right.}$$
$$=4-\sqrt{3}$$

## 2015 May June Paper 22 Q3

Do not use a calculator in this question.

The diagram shows the right-angled triangle $$A B C$$, where $$A B=(6+3 \sqrt{5}) cm$$ and angle $$B=90^{\circ} .$$ The area of this triangle is $$\left(\frac{36+15 \sqrt{5}}{2}\right) cm ^{2}$$.

1. Find the length of the side $$B C$$ in the form $$(a+b \sqrt{5}) cm$$, where $$a$$ and $$b$$ are integers.
2. Find $$(A C)^{2}$$ in the form $$(c+d \sqrt{5}) cm ^{2}$$, where $$c$$ and $$d$$ are integers.

Available soon

## 2016 Oct Nov Paper 23

In this question all lengths are in centimetres.

In the triangle $$A B C$$ shown above, $$A C=\sqrt{3}+1, B C=\sqrt{3}-1$$ and angle $$A C B=60^{\circ}$$.

1. Without using a calculator, show that the length of $$A B=\sqrt{6}$$.
2. Show that angle $$C A B=15^{\circ}$$.
3. Without using a calculator, find the area of triangle $$A B C$$.

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