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If 64:x::x:169, and x>0, find the value of x.
104
108
112
96
104
In a continued proportion a:x::x:b, the middle term x is the geometric mean of the extreme terms a and b, which is simply the square root of their product.
A continued proportion is given as 64: x:: x: 169 with the condition that x > 0.
x=a├ЧbтАЛ
In a continued proportion a:x::x:b, the middle term x is the geometric mean of the extreme terms a and b, which is simply the square root of their product.
Students often miscalculate by forgetting that the mean proportional x must be positive, or they may incorrectly perform the cross-multiplication leading to x2=64├Ч169 and then fail to calculate the square root accurately.
Set up the proportion equation
Given the proportion 64:x::x:169, we can express it as a ratio of fractions: x64тАЛ=169xтАЛ.
x64тАЛ=169xтАЛ
Solve for x squared
Cross-multiply the terms to isolate x2: x├Чx=64├Ч169, which gives x2=10816.
x2=64├Ч169=10816
Calculate the value of x
Take the square root of both sides. Since x>0, we have x=64тАЛ├Ч169тАЛ=8├Ч13.
x=8├Ч13=104
A is correct because the mean proportional x is calculated as 64├Ч169тАЛ=8├Ч13=104.
This concept of mean proportion is identical to finding the geometric mean in sequences and series, which frequently appears in advanced geometry problems involving similar triangles.
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