Calc
posted by Ben .
Profit=p(p^2+33p+9)9(p^2+33p+9)+100
How do I simplify this so I can take the derivative
For Further Reading
* Calc  Michael, Sunday, November 25, 2007 at 4:17pm
Profit=p(p^2+33p+9)9(p^2+33p+9)+100
p(p^2 + 33p + 9)
Just distribute the p in.
9(p^2 + 33p + 9)
Find the derivative of the (p^2 + 33p + 9) and then multiply it by 9. You can distribute the 9 in at the beginning, but it's not necessary.
+100
The derivative of a consonant is 0.
I hope that helps. If you have any questions, let me know.
o Calc  Ben, Sunday, November 25, 2007 at 6:55pm
I got 3p^2+84p+298
Is this right?
+ Calc  Ben, Sunday, November 25, 2007 at 6:58pm
I made a mistake it should be 296 but I do not think this is right because I know I am supposed to factor this and I cannot get it to factor.
o Calc  Ben, Sunday, November 25, 2007 at 7:08pm
That is wrong too Now I got
3p^2+84p288
Is this right, I cannot figure out how to factor it though
+ Calc  Michael, Sunday, November 25, 2007 at 7:31pm
Yes, that's correct. To factor, you can take out a common number. (Take out a negative to make it easier to work with, too.) Try that, and see what you get.
* Calc  Ben, Sunday, November 25, 2007 at 7:36pm
I got 3(p^228p+96)
Now How do I factor this further I need to eventually set it equal to zero
o Calc  Michael, Sunday, November 25, 2007 at 7:42pm
Don't worry about setting it equal to 0. Since we're factoring, it is equal to 0. (You can write that = 0 in your work if you want.)
That's difficult to continue factoring. Here's a hint: 4 x 24 is 96.
+ Calc  Ben, Sunday, November 25, 2007 at 7:58pm
Wait, the other one is wrong, it is
3(p24)(p4).
So the max possible weekly profit is $24 dollars???
Now how do I determine the max possible weekly profit and be certain the profit is maximized?
* Calc  Ben, Sunday, November 25, 2007 at 7:54pm
So is it 3(p24)(p+4)
I have to find which will give me my largest profit so I need to set them equal to zero. That would then be $24, right?
How do I determine max possible weekly profit?
o Calc  Michael, Sunday, November 25, 2007 at 8:02pm
Don't forget that factoring gives you the xintercepts of an equation. Maxima are the highest and lowest yvalues.
Have you learned the First Derivative Test?
+ Calc  Ben, Sunday, November 25, 2007 at 8:06pm
No, But should I charge $24 to get the largest weekly profit?

3(p24)(p+4) = 0
Solve that for p. You should get two answers.
Then, plug each of those into your original PROFIT equation to get the max. 
I got 24 and 4.

p  24 = 0, p + 4 = 0
p = 24, p = 4
Then, plug each of those into your original PROFIT equation to get the max. 
With 4 I got 508 and with 24 I got 6893
What did I do wrong? This makes no sense 
Profit=p(p^2+33p+9)9(p^2+33p+9)+100
If you plugged it into that, check your algebra. 
I got 20755 when I plugged 24 in and 685 when I plugged 4 in, is this correct?

No, that's still not right. You should have plugged in 24 and NEGATIVE 4.
Anyway, forget the 4. p=24 will give you the greatest profit. All you have to do is plug and chug. 
But how is it 4, I am confused

We were solving 3(p24)(p+4) = 0.
We do that by setting each parenthesis equal to 0.
p + 4 = 0
Subtract 4 from both sides.
p = 4 
okay sorry. How can I be certain that the profit is being maximized?

I'm not sure. Try the general justification for a maximum:
The derivative is changing from negative to positive.
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