Minimum Problem

[Jason76]

Below is the solution for a similar problem:



Here is my problem:

 

[Jason76]

\(\displaystyle xy = 37.5,000,000 \)

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.

 

[Jason76]

The above work is (on the 2nd post of the thread) is still wrong on homework.

 

[HallsofIvy]

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?

 

[Jason76]

This problem was solved here: http://mathhelpforum.com/calculus/235621-minimum-problem.html