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Question 210894This question is from textbook COLLEGE ALGEBRA
: Pls. Help can't figure it out
The law of divine proportion, according to early Egyptians and Greek, states, “the most beautiful division of a line segment into two parts is such that the ratio of the longer part to the total length is equal to the ratio of the length of the shorter part to that of the longer part”
a.) If we consider the length of the line to be one unit and call t the length of the longer of the two parts, Find t.
b.) What is the proportionality constant for this ratio?
thank you very much
This question is from textbook COLLEGE ALGEBRA
Found 2 solutions by Theo, stanbon:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! let x = shorter part
let y = longer part
x + y = 1 (given)
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x is to y as y is to 1
ratio is:
x/y = y/1
cross multiply to get:
y^2 = x
which is the same as:
x = y^2
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since x + y = 1, we can replace x with y^2 to get:
y^2 + y = 1
we can complete the square to get:
(y+1)^2 = 1.25
we take the square root of both sides to get:
y+1 = +/- sqrt(1.25)
we subtract 1 from both sides to get:
y = -1 +/- sqrt(1.25)
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solving for y gets:
y = .618033989
or:
y = -1.618033989
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since y can't be negative, the only possible solution is:
y = .618033989
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since x + y = 1, this means that:
x = .381966011
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we have:
x = .381966011
y = .618033989
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x is the shorter segment
y is the longer segment
x/y = .381966011/618033989 = .618033989
y/1 = .618033989
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the ratio of shorter segment to longer segment is the same as the ratio of longer segment to the total length.
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our answer is that the longer segment = .618033989
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constant of proportionality is something else.
these are directly proportional, so the formula for k would be:
y = k*x
substituting known values for x and y, we get:
.618033989 = k*.381966011
solving for k, we get:
k = .618033989/.381966011 = 1.618033989
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if you know x, you can find y by multiplying by 1.618033989
.381966011 * 1.618033989 = .618033989
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if you know y, you can find x by dividing by 1.618033989
.618033989 / 1.618033989 = .381966011
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if you have problems with any of this, just send me an email with your questions and i'll reply as soon as i can.
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Answer by stanbon(75887)  (Show Source):
You can put this solution on YOUR website! The law of divine proportion, according to early Egyptians and Greek, states, “the most beautiful division of a line segment into two parts is such that the ratio of the longer part to the total length is equal to the ratio of the length of the shorter part to that of the longer part”
a.) If we consider the length of the line to be one unit and call t the length of the longer of the two parts, Find t.
Draw a line segment and mark a point that is not in the middle.
Label the larger part as "a" and the smaller as "1-a"
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Write the proportion as described above.
a/1 = (1-a)/a
Cross-multiply and solve for "a":
a^2 = 1-a
a^2+a-a = 0
a = [-1+-sqrt(1-4*-1)]/2
Positive solution:
a = [-1+sqrt(5)]/2 = 0.618
Then b = 1-0.618 = 0.3817
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b.) What is the proportionality constant for this ratio?
a/b = 1.618 which is called the Golden Proportion.
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Cheers,
Stan H.
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