Question 2077: Hi, have no idea how to solve this: A formula of dimonds price is C=am^2, m - is a dimond's mass, a - is a constant number independent of diamond's mass. So the dimond is cut into two pieces.
1. Find a ratio of the masses of those two pieces, when the sum of the prices of these two pieces equals 5/9 of a whole uncut dimond price.
2. What is a ratio of the masses of those two pieces, when the sum of the prices of these two pieces is lowest?
:)
Answer by khwang(438)   (Show Source):
You can put this solution on YOUR website! 1. Sol:Assume the first piece is rm, and the 2nd piece is (1-r)m,
where 0<= r <=1.
The sum of two prices = a(rm)^2 + a((1-r)m)^2 = a(2r^2 -2r+1)m^2
= 5/9 am^2.
So, 2r^2 -2r+1 = 5/9, or 9r^2 -9r + 2 = 0.
By factoring (3r -1)(3r -2) = 0, we have r=1/3 or 2/3.
[Note: In my opinion, if you can factor out directly,
don't use quadratic formula.]
Hence,the ratio of the two pieces is r:(1-r) = 1/3:2/3 =1:2
(or 2:1).
2.Use complete square
f(r) = 2r^2 -2r+1 = 2(r^2 -r+1/4) +1/2 = 1/2 + 2(r-1/2)^2 => 1/2.
Since 2(r-1/2)^2 >= 0, when r = 1/2, f(r) has minimum value 1/2.
That is, if the ratio = r:(1-r) = 1:1, the the sum of the prices of
these two pieces is lowest.
Kenny
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