Asked 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+84p-288

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^2-28p+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(p-24)(p-4).
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(p-24)(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 x-intercepts of an equation. Maxima are the highest and lowest y-values.

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?
Answered by Michael
-3(p-24)(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.
Answered by Ben
I got 24 and 4.
Answered by Michael
p - 24 = 0, p + 4 = 0
p = 24, p = -4

Then, plug each of those into your original PROFIT equation to get the max.
Answered by Ben
With -4 I got 508 and with 24 I got -6893
What did I do wrong? This makes no sense
Answered by Michael
Profit=p(-p^2+33p+9)-9(-p^2+33p+9)+100

If you plugged it into that, check your algebra.
Answered by Ben
I got 20755 when I plugged 24 in and -685 when I plugged 4 in, is this correct?
Answered by Michael
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.
Answered by Ben
But how is it -4, I am confused
Answered by Michael
We were solving -3(p-24)(p+4) = 0.

We do that by setting each parenthesis equal to 0.

p + 4 = 0
Subtract 4 from both sides.

p = -4
Answered by Ben
okay sorry. How can I be certain that the profit is being maximized?
Answered by Michael
I'm not sure. Try the general justification for a maximum:

The derivative is changing from negative to positive.
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