A golden rectangle is 32cm long. The ratio of length to width is (1 + square root of 5) : 2. What is the width of the rectangle in simplest radical form?

Width = 16 (√5) - 1

Step-by-step explanation:

We are told that the golden rectangle is 32 cm long.

Thus, length = 32 cm

We are also told that the ratio of the length to the width is; (1 + √5) : 2

Thus;

If a length of (1 + √5) yields a width of 2

Then, a length of 32 cm would yield a width of; (32 x 2) / (1 + √5)

So corresponding width = 64 / (1 + √5)

We want to reduce this width to it's simplest radical form which means the denominator should have no square root.

Thus, let's multiply top and bottom by (1 - √5);

Width = 64 x (1 - √5) / [ (1 + √5) x (1 - √5) ]

Width = 64 (1 - √5) / (1 - 5)

Width = 64 (1 - √5) / (-4)

Width = - 16 (1 - √5)

Width = 16 (√5 - 1)

Width = 16√5 - 1