Question

Precalculus help?
The total revenue R earned (in dollars) from producing a gift box of candles is given by: R(p)=-10 + 800p.
Find the unit price that will yield a maximum revenue. What is the maximum revenue? Explain your results.

Answer 1

are u sure about
\[ \large R(p)=-10+800p \]
this is a line. it grows and grows as \(p\) does.

Answer 2

Oops, I'm sorry. I mean, R(p) + -10p^2 +800p.

Answer 3

do u know derivatives?

Answer 4

Not familiar with that, no. I could, but maybe I just don't know the term.

Answer 5

never mind. let's complete the square then.
do u know this??

Answer 6

Divide 800 by 2, square it? 1,600?

Answer 7

u have
\[ \large R= R(p)=-10p^2+800p=-10(p^2-80p) \]

Answer 8

Oh okay. But where do I go from there? Sorry, this whole thing is pretty confusing to me.

Answer 9

no problem.
\[ \large R-10\left(\frac{80}{2}\right)^2=-10\left[p^2-80p+
\left(\frac{80}{2}\right)^2\right]=-10(p-40)^2 \]

Answer 10

Where did that 2 come from?

Answer 11

to complete the square u divide 80 by 2 and square it.
right??

Answer 12

Amanda??

Answer 13

Okay. But what happened to your p^2 and -80p in the last step?

Answer 14

what do u get if u expand
\[ \large (p-40)^2 \]

Answer 15

Oh alright. Well so now if I have -10(p-40)^2, what's my next step?

Answer 16

I'm sorry, I feel terrible asking so many questions.

Answer 17

don't worry. this site is for that precisely. and us, willing participants, are here to help.

now u have
\[ \large R-16000=-10(p-40)^2 \]
this is a parabola that opens downwards and has vertex (40,16000)

Answer 18

Does it really go any further than adding 16,000 to both sides? I know that would put it in the perfect graphing form. The maximum seems like it would be the vertex, am I right?

Answer 19

yes. the price asked for is k=40 and the maximum revenue is R=16,000

Answer 20

sorry p=40.

Answer 21

Thank you! And I'm also given values for prices and told to find the revenue. The prices are 20, 25, 30. I assume I do pretty much the same exact thing, except with values plugged in for p?

Answer 22

yes. just plug the values of p you are given into the formula.

Answer 23

Alright. Thanks so much, you've been a big help, I really appreciate it. :)

Answer 24

for instance
\[ \large R(20)=-10\cdot(20)^2+800\cdot20= \]
i leave this to you.

and u r welcome.