Minimum Problem
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\(\displaystyle xy = 37.5,000,000 \)
minimize \(\displaystyle C = 3x + 2y\)
solve \(\displaystyle xy = 37.5,000,000\) for \(\displaystyle y\)
\(\displaystyle y = \dfrac{37.5,000,000}{x}\)
\(\displaystyle C = C(x) = 3x + 2(37.5,000,000)x^{-1}\)
\(\displaystyle C'(x) = 3 - (75,000,000)x^{-2}\)
\(\displaystyle 3 - (75,000,000)x^{-2} = 0\)
\(\displaystyle -75,000,000x^{-2} = -3\)
\(\displaystyle x^{-2} = \dfrac{-3}{-75,000,000} \)
\(\displaystyle (x^{-2})^{-1} = (\dfrac{-3}{-75,000,000})^{-1} \)
\(\displaystyle x^{2} = \dfrac{-75,000,000}{-3}\)
\(\displaystyle x^{2} = 25,000,000\)
\(\displaystyle (x^{2})^{1/2} = (25,000,000)^{1/2}\)
\(\displaystyle x = \pm 5000 \)
\(\displaystyle C(5000) = 3(5000) + \dfrac{2(37.5,000,000)}{5000} = 30000\)
\(\displaystyle C(5000) = 3(-5000) + \dfrac{2(37.5,000,000)}{-5000} = -30000\)
\(\displaystyle y = \dfrac{37.5,000,000}{5000} = 7500\) ft
All this stuff is wrong on homework.
minimize \(\displaystyle C = 3x + 2y\)
solve \(\displaystyle xy = 37.5,000,000\) for \(\displaystyle y\)
\(\displaystyle y = \dfrac{37.5,000,000}{x}\)
\(\displaystyle C = C(x) = 3x + 2(37.5,000,000)x^{-1}\)
\(\displaystyle C'(x) = 3 - (75,000,000)x^{-2}\)
\(\displaystyle 3 - (75,000,000)x^{-2} = 0\)
\(\displaystyle -75,000,000x^{-2} = -3\)
\(\displaystyle x^{-2} = \dfrac{-3}{-75,000,000} \)
\(\displaystyle (x^{-2})^{-1} = (\dfrac{-3}{-75,000,000})^{-1} \)
\(\displaystyle x^{2} = \dfrac{-75,000,000}{-3}\)
\(\displaystyle x^{2} = 25,000,000\)
\(\displaystyle (x^{2})^{1/2} = (25,000,000)^{1/2}\)
\(\displaystyle x = \pm 5000 \)
\(\displaystyle C(5000) = 3(5000) + \dfrac{2(37.5,000,000)}{5000} = 30000\)
\(\displaystyle C(5000) = 3(-5000) + \dfrac{2(37.5,000,000)}{-5000} = -30000\)
\(\displaystyle y = \dfrac{37.5,000,000}{5000} = 7500\) ft
All this stuff is wrong on homework.
Last edited:
HallsofIvy
Elite Member
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- Jan 27, 2012
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First, please do not write "37.5 million" as "37.5,000,000". The correct notation is 37,500,000. Surely you know that!
Also, there is no reason to keep writing "\(\displaystyle \frac{-3}{-75,000,000}\). Go ahead and get rid of the cancelling "-"s immediately.
But you did the calculations correctly, yes, x= 5000 feet. Then \(\displaystyle y= \frac{37,500,000}{5000}= 7500\) feet.
I don't see anywhere that you wrote your answer "x= 5000 ft, y= 7500 ft".
Also, I don't see why you calculated the length of the fencing for x= 5000 and certainly not why you calculated it for "x= -5000". Surely you know that the length of a fence cannot be negative?
Also, there is no reason to keep writing "\(\displaystyle \frac{-3}{-75,000,000}\). Go ahead and get rid of the cancelling "-"s immediately.
But you did the calculations correctly, yes, x= 5000 feet. Then \(\displaystyle y= \frac{37,500,000}{5000}= 7500\) feet.
I don't see anywhere that you wrote your answer "x= 5000 ft, y= 7500 ft".
Also, I don't see why you calculated the length of the fencing for x= 5000 and certainly not why you calculated it for "x= -5000". Surely you know that the length of a fence cannot be negative?
This problem was solved here: http://mathhelpforum.com/calculus/235621-minimum-problem.html