The length of the bracket after 1 iteration is 100%.
Let f(x) = x³-15x² + 50x
a = 0, b=10.
Let the golden ratio be
GR=√5-1 / 2=0.618
Now, d = GR (b-a)
= 0.618 (10-0) = 6·18
[tex]x_{1}[/tex]= a+d = 0 + 6·18 = 6.18
[tex]x_{2}[/tex] = b-d = 10- 6·18 = 3.82
f([tex]x_{1}[/tex]) = (6·18)³ - 15 (6·18)² + 50(6·18) = -27.887
f([tex]x_{2}[/tex]) = (3.82)³-15(3.82)²+50(3.82)=27.857
so, f([tex]x_{2}[/tex]) > f([tex]x_{1}[/tex]) new interval is [a,x,]
i.e., maximum lies in [0, 6·18]
so, Xmax = [tex]x_{2}[/tex]= 382
Uncertainty in measurement will be,
∈= [tex]\frac{1-GR) (b-a)}{Xopt}[/tex] =[tex]\frac{((1-0.618) X (10-0)}{382}[/tex] × 100%
∈ = 100%
In arithmetic, quantities are within the golden ratio if their ratio is the same as the ratio in their sum to the larger of the 2 portions. Expressed algebraically, for quantities a and b with [tex]a > b > 0[/tex], [tex]\frac{(a+b)}{a}[/tex] = [tex]\frac{a}{b}[/tex]= φ
The golden ratio turned into known as the acute and suggest ratio by way of Euclid, and the divine proportion using Luca Pacioli, and also is going through numerous other names. The golden ratio seems in some patterns in nature, such as the spiral association of leaves and different elements of plants.
A few twentieth-century artists and architects, together with Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio, believing them to be aesthetically appealing. these uses often appear in the form of a golden rectangle.
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