A rectangle has an area of 10\text{ cm}^2 and a length of (\sqrt{7} + \sqrt{2})\text{ cm}.… — JAMB UTME

A rectangle has an area of $10\text{ cm}^2$ and a length of $(\sqrt{7} + \sqrt{2})\text{ cm}$ . What is the width of the rectangle in simplified surd form?

$(2\sqrt{7} - 2\sqrt{2})\text{ cm}$ Answer
B
$(2\sqrt{7} + 2\sqrt{2})\text{ cm}$
C
$2\sqrt{5}\text{ cm}$
D
$\frac{10\sqrt{7} - 10\sqrt{2}}{9}\text{ cm}$

Answer

The width of the rectangle in simplified surd form is

(2\sqrt{7} - 2\sqrt{2})\text{ cm}

The area formula for a rectangle gives

\text{width} = \frac{10}{\sqrt{7} + \sqrt{2}}

. Multiplying both numerator and denominator by the conjugate

(\sqrt{7} - \sqrt{2})

produces

\frac{10(\sqrt{7} - \sqrt{2})}{7 - 2} = \frac{10(\sqrt{7} - \sqrt{2})}{5} = 2\sqrt{7} - 2\sqrt{2}

Step-by-Step Solution

Set up the formula for the width of the rectangle

\text{Width} = \frac{\text{Area}}{\text{Length}} = \frac{10}{\sqrt{7} + \sqrt{2}}

The area of a rectangle is length multiplied by width.

Rationalize the denominator by multiplying top and bottom by the conjugate

(\sqrt{7} - \sqrt{2})

\text{Width} = \frac{10(\sqrt{7} - \sqrt{2})}{(\sqrt{7} + \sqrt{2})(\sqrt{7} - \sqrt{2})}

Multiplying by the conjugate creates a difference of squares in the denominator, eliminating radicals.

Simplify the denominator using

(a+b)(a-b) = a^2 - b^2

(\sqrt{7})^2 - (\sqrt{2})^2 = 7 - 2 = 5

Squaring a square root yields the underlying rational number.

Divide the numerator by the simplified denominator

\frac{10(\sqrt{7} - \sqrt{2})}{5} = 2(\sqrt{7} - \sqrt{2}) = 2\sqrt{7} - 2\sqrt{2}

Dividing

10

5

gives

2

, which is then distributed across the terms inside the parentheses.

Key Concept

Rationalization of Denominators with Binomial Surds

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