anonymous
  • anonymous
HELP! The following function shows the relationship between the selling prices, and profit P(s), in dollars, for a company: P(s) = -20s2 + 1,400s - 12,000 Which statement best describes the intervals where the company's profit increases, decreases, or records a maximum? a) It is least when the selling price is $30. b) It is greatest when the selling price is $30. c) It decreases when the selling price increases from $10 to $35. d) It increases when the selling price increases from $10 to $35. Explanation for answer would be greatly appreciated. Thanks.
Mathematics
  • Stacey Warren - Expert brainly.com
Hey! We 've verified this expert answer for you, click below to unlock the details :)
SOLVED
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
jamiebookeater
  • jamiebookeater
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
Anaise
  • Anaise
hi
SolomonZelman
  • SolomonZelman
Calc 1?
SolomonZelman
  • SolomonZelman
Or just a vertex problem?

Looking for something else?

Not the answer you are looking for? Search for more explanations.

More answers

anonymous
  • anonymous
I wish...but nah Algebra I
SolomonZelman
  • SolomonZelman
Well, we don't need calculus really, since it is a parabola..
SolomonZelman
  • SolomonZelman
\(\color{#000000 }{ \displaystyle p(s)=-20s^2+1400s-12000 }\)
SolomonZelman
  • SolomonZelman
Can you factor the left side out of -20?
anonymous
  • anonymous
Sure, one sec.
anonymous
  • anonymous
-20(s^2-700s+600)
SolomonZelman
  • SolomonZelman
wait the middle
SolomonZelman
  • SolomonZelman
middle term is wrong.
anonymous
  • anonymous
Sorry, -20 (s^2 + 70s + 600)
anonymous
  • anonymous
Wait isn't that wrong?
anonymous
  • anonymous
Oh man i factored 14000 instead, one sec
SolomonZelman
  • SolomonZelman
\(\color{#000000 }{ \displaystyle -20 (s^2 - 70s + 600) }\)
anonymous
  • anonymous
-20 (s-10) (s-60)
SolomonZelman
  • SolomonZelman
no need for complete factorization.
SolomonZelman
  • SolomonZelman
\(\color{#000000 }{ \displaystyle p(s)=-20 (s^2 - 70s + 600) }\) is what you need at this point.
anonymous
  • anonymous
Okay i got that
SolomonZelman
  • SolomonZelman
you need to complete the square... What would you add \(\color{#000000 }{ \displaystyle s^2 - 70s + ~? }\) to make a perfect square trinomial?
SolomonZelman
  • SolomonZelman
Note that, \(\color{#000000 }{ \displaystyle x^2+2ax+a^2\color{grey}{~~~~~=(x+a)^2}}\) is something that you want to obtain
SolomonZelman
  • SolomonZelman
so the 2a corresponds (in our case), to -70, right?
anonymous
  • anonymous
It's not a perfect square, right?
SolomonZelman
  • SolomonZelman
the x²+2ax+a² is a perf. sq.
SolomonZelman
  • SolomonZelman
your polynomial inside the parenthesis is not,,,
anonymous
  • anonymous
Yea, in the parenthesis, the first term and last term are perfect squares, but the middle one isn't, right?
SolomonZelman
  • SolomonZelman
but if you tell me the missing number for \(\color{#000000 }{ \displaystyle s^2 - 70s +~{\rm what?} }\) then I can show you a trick.
SolomonZelman
  • SolomonZelman
"what?" will make the s²-70s a perfect square trinomial when added?
anonymous
  • anonymous
I don't know
SolomonZelman
  • SolomonZelman
\(s^2+2as+a^2\) is the form you want
SolomonZelman
  • SolomonZelman
you have \(s^2-70s\)
SolomonZelman
  • SolomonZelman
So -70s, corresponds to the "2as" peace, right?
SolomonZelman
  • SolomonZelman
So, what is the "a" in our case?
anonymous
  • anonymous
-35?
SolomonZelman
  • SolomonZelman
yup
SolomonZelman
  • SolomonZelman
and "a²" is what?
anonymous
  • anonymous
YAY!
SolomonZelman
  • SolomonZelman
ok, next question i asked...
anonymous
  • anonymous
(-35)^2
SolomonZelman
  • SolomonZelman
yes, and that would be?
anonymous
  • anonymous
1225
SolomonZelman
  • SolomonZelman
yup
SolomonZelman
  • SolomonZelman
So, I will show you the trick now...
SolomonZelman
  • SolomonZelman
\(\color{#000000 }{ \displaystyle p(s)=-20 (s^2 - 70s + 600) }\) You already have 600 in parenthesis, and you need to make that a 1225. (For this to be a perfect square trinomial) \(\color{#000000 }{ \displaystyle 600+x=1225 }\) \(\color{#000000 }{ \displaystyle x=625 }\) But, you can't just add 625, you will have to use something that I refer to as the "magic zero", and you will see what I am talking about NOW... \(\color{#000000 }{ \displaystyle p(s)=-20 (s^2 - 70s + 600+625-625) }\) Now, we would like to get rid of -625 in parenthesis, but we can't just erase it. We will take it out, by multiplying times -20.
SolomonZelman
  • SolomonZelman
\(\color{#000000 }{ \displaystyle p(s)=-20 (s^2 - 70s + 600+625) +(-20)(-625)}\) \(\color{#000000 }{ \displaystyle p(s)=-20 (s^2 - 70s + 1225) +1250}\) and recall the form, \(s^2+2as+a^2=(s+a)^2\) We have already clarified that: a=-35 a²=1225 2a=-70 \(\color{#000000 }{ \displaystyle p(s)=-20 (s^2 - 2(35)s +(-35)^2) +1250}\)
SolomonZelman
  • SolomonZelman
And we apply the form now! \(\color{#000000 }{ \displaystyle p(s)=-20 (s - 35)^2 +1250}\)
SolomonZelman
  • SolomonZelman
when you digest this info, let me know...
anonymous
  • anonymous
One moment, still digesting
anonymous
  • anonymous
I got it. What's next?
SolomonZelman
  • SolomonZelman
\(\color{#000000 }{ \displaystyle p(s)=-20 (s - 35)^2 +1250}\) can you tell me if this parabola opens up or down (and why)?
anonymous
  • anonymous
down? i think..
SolomonZelman
  • SolomonZelman
yes, and why?
anonymous
  • anonymous
negative coefficient?
SolomonZelman
  • SolomonZelman
yes, the leading coefficient is negative... fabulous!
SolomonZelman
  • SolomonZelman
Since the parabola opens down, the vertex is the maximum point of the parabola. (The other points are all lower)
SolomonZelman
  • SolomonZelman
So you need to find the vertex, and for this there is a rule...
SolomonZelman
  • SolomonZelman
\(\color{#000000 }{ \displaystyle y(x)=a(x-k)^2+h }\) will have a vertex of \(\color{#000000 }{ \displaystyle (h,k) }\)
anonymous
  • anonymous
(1250,-35)?
anonymous
  • anonymous
Sorry (1250, 35)
SolomonZelman
  • SolomonZelman
not k,h the other way around
SolomonZelman
  • SolomonZelman
my fault
SolomonZelman
  • SolomonZelman
\(\color{#000000 }{ \displaystyle y(x)=a(x-h)^2+k }\) will have a vertex of \(\color{#000000 }{ \displaystyle (h,k) }\)
SolomonZelman
  • SolomonZelman
that is the form
anonymous
  • anonymous
(35, 1250)
SolomonZelman
  • SolomonZelman
yes, correct, and I apologize for my mistake.
anonymous
  • anonymous
No worries, you have been amazing help!
anonymous
  • anonymous
And guidance
SolomonZelman
  • SolomonZelman
And the vertex of (35,1225) indicates that the maximum profit is 1225, and it occurs when you sell 35 dollars per item.
SolomonZelman
  • SolomonZelman
Which statement best describes the intervals where the company's profit increases, decreases, or records a maximum? a) It is least when the selling price is $30. b) It is greatest when the selling price is $30. c) It decreases when the selling price increases from $10 to $35. d) It increases when the selling price increases from $10 to $35.
anonymous
  • anonymous
d?
SolomonZelman
  • SolomonZelman
you are close.
anonymous
  • anonymous
c?
SolomonZelman
  • SolomonZelman
can you sketch (somehow) a parabola that opens down please?
SolomonZelman
  • SolomonZelman
any picture that looks like opening-down parabola would suffice.
anonymous
  • anonymous
|dw:1450410288830:dw|
SolomonZelman
  • SolomonZelman
good
SolomonZelman
  • SolomonZelman
|dw:1450410304542:dw|
SolomonZelman
  • SolomonZelman
|dw:1450410339366:dw|
SolomonZelman
  • SolomonZelman
is the slope of this line positive or negative?
anonymous
  • anonymous
positive
SolomonZelman
  • SolomonZelman
good
SolomonZelman
  • SolomonZelman
|dw:1450410402632:dw|
SolomonZelman
  • SolomonZelman
is the slope of this (red) tangent line positive or negative?
anonymous
  • anonymous
positive
SolomonZelman
  • SolomonZelman
ok
SolomonZelman
  • SolomonZelman
|dw:1450410507209:dw|
SolomonZelman
  • SolomonZelman
all of these Lines (A, B, C and L) are tangent to the parabola
SolomonZelman
  • SolomonZelman
Which lines have negative slopes and which have positive slopes?
anonymous
  • anonymous
A and C have positive, B and L have negative
SolomonZelman
  • SolomonZelman
awesome!
SolomonZelman
  • SolomonZelman
negative slope indicates that the price is increasing, right?
SolomonZelman
  • SolomonZelman
yes or no?
anonymous
  • anonymous
No..?
SolomonZelman
  • SolomonZelman
Yes,
SolomonZelman
  • SolomonZelman
the price is decreasing if slope is negative
SolomonZelman
  • SolomonZelman
And is increasing if the slope is positive
SolomonZelman
  • SolomonZelman
|dw:1450410804794:dw|
SolomonZelman
  • SolomonZelman
So does the price increase or decrease from s=10 and up to s=35?
anonymous
  • anonymous
increase
SolomonZelman
  • SolomonZelman
yes, and thus you know the answer...
anonymous
  • anonymous
Wait is it c or d?
anonymous
  • anonymous
d, right?
SolomonZelman
  • SolomonZelman
it increases up to 35....
SolomonZelman
  • SolomonZelman
D
anonymous
  • anonymous
Gotcha, thanks!!
SolomonZelman
  • SolomonZelman
yw

Looking for something else?

Not the answer you are looking for? Search for more explanations.