I am reposting this because I still don't understand it.

I need help with this problem. this is what I got so far:

P(x)=12x-(x^2/500)-680-4x-0.01x^2

P'(x)=12-0.004x-4-0.02x

P'(x)= -0.024x-8

x=8/0.024

My book says x=8/0.024 which somehow become 1000/3. They are the same when solved but how did they come up with 1000/3?

Someone answered and said to multiply numerator and denomerator by 1000/8. But WHY do I need to this? And how do I know what ot multiply by? I don't get it.

we usually don't have decimals in fractions, either we use a fraction or we use a decimal

so
8/0.024
= 8/0.024 * (1000/1000)
= 8000/24 , now reduce by a factor of 8
= 1000/3

I believe it was drwls who suggested to multiply numerator and denomerator by 1000/8.
He merely combined both of my above steps into one operation.

(I am somewhat surprised that, taking Calculus, you are confused by such a trivial operation.)

Thanks I get it now. I just couldn't figure out why they got 1000/3 and I got 8/0.024. I don't know why I didn't see that, just simple algebra.

To understand how the value of x in the equation x=8/0.024 becomes 1000/3, let's step through the process.

In your question, you correctly found the derivative of the function P(x) as P'(x) = -0.024x - 8. To solve for x, you set P'(x) equal to 0, as finding the values of x for which the derivative is zero can help identify critical points.

So, we have -0.024x - 8 = 0. To isolate x, we need to get rid of the -8. We can do this by adding 8 to both sides of the equation:

-0.024x = 8.

Now, to solve for x, we need to divide both sides of the equation by -0.024:

x = 8/(-0.024).

Dividing 8 by -0.024 gives us -333.3333. However, this doesn't match the value 1000/3 mentioned in your book. To reconcile this difference, we need to multiply the numerator and denominator of x = 8/(-0.024) by the same value, so that the value doesn't change.

In this case, we multiply by 1000/8 because it will give us a common multiple of 1000 in the numerator and a common factor of 8 in the denominator. This allows us to express the fraction in a different form without changing its value.

Multiplying both the numerator and denominator of x = 8/(-0.024) by 1000/8, we get:

x = (8 * 1000) / (-0.024 * 8).

Now, simplifying the expression:

x = 8000 / (-0.192).

Dividing 8000 by -0.192 yields -41666.6667.

So, the value of x is indeed -41666.6667, which can be further simplified to -41666 and two-thirds, or -41666 2/3. This is equivalent to the fraction 1000/3.

Therefore, the solution x = 8/0.024 can be written as x = 1000/3.

To summarize, multiplying x = 8/(-0.024) by 1000/8 ensures that the value of x remains the same while allowing us to arrive at a more simplified form, which is consistent with your book's answer of x = 1000/3.