A golden rectangle is 32 cm long. The ratio of length to width is (1 + sq rt 5)/2. What is the width of the rectangle in simplest radical form?
Please solve and explain. Thanks.
This is a horrible explanation that is hard to understand.
I agree with you, the formatting and steps seem to be off.
To find the width of the golden rectangle, we can use the formula for the ratio of length to width:
Ratio = Length / Width
In this case, the ratio given is (1 + sqrt(5))/2, and the length is given as 32 cm. Let's substitute these values into the formula:
(1 + sqrt(5))/2 = 32 / Width
To find the width, we can rearrange the equation:
Width = 32 / ((1 + sqrt(5))/2)
To simplify this expression, we can multiply both the numerator and denominator by 2:
Width = (32 * 2) / (1 + sqrt(5))
Width = 64 / (1 + sqrt(5))
Next, to simplify the expression further, we can rationalize the denominator by multiplying the numerator and denominator by the conjugate of (1 + sqrt(5)). The conjugate of (1 + sqrt(5)) is (1 - sqrt(5)):
Width = (64 / (1 + sqrt(5))) * ((1 - sqrt(5))/(1 - sqrt(5)))
Using the distributive property, we can multiply the numerators and the denominators:
Width = (64 * (1 - sqrt(5))) / ((1 + sqrt(5)) * (1 - sqrt(5)))
Simplifying the numerator:
Width = (64 - 64 * sqrt(5)) / (1 - 5)
Width = (64 - 64 * sqrt(5)) / (-4)
Finally, simplifying the expression further:
Width = -16 + 16 * sqrt(5)
So, the width of the golden rectangle, in simplest radical form, is -16 + 16 * sqrt(5).
L = Length
W = Width
L / W = ( 1 + sqrt 5 ) / 2
Reciprocally :
W / L = 1 / [ ( 1 + sqrt 5 ) / 2 ]
W / L = 2 / [ ( 1 + sqrt 5 ) ] Multiply both sides by L
W = 2 L / 2 / [ ( 1 + sqrt 5 ) ]
W = 2 * 32 / [ ( 1 + sqrt 5 ) ]
W = 64 / [ ( 1 + sqrt 5 ) ]