Question

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.

Answer 1

hi

Answer 2

Calc 1?

Answer 3

Or just a vertex problem?

Answer 4

I wish...but nah Algebra I

Answer 5

Well, we don't need calculus really, since it is a parabola..

Answer 6

\(\color{#000000 }{ \displaystyle p(s)=-20s^2+1400s-12000 }\)

Answer 7

Can you factor the left side out of -20?

Answer 8

Sure, one sec.

Answer 9

-20(s^2-700s+600)

Answer 10

wait the middle

Answer 11

middle term is wrong.

Answer 12

Sorry, -20 (s^2 + 70s + 600)

Answer 13

Wait isn't that wrong?

Answer 14

Oh man i factored 14000 instead, one sec

Answer 15

\(\color{#000000 }{ \displaystyle -20 (s^2 - 70s + 600) }\)

Answer 16

-20 (s-10) (s-60)

Answer 17

no need for complete factorization.

Answer 18

\(\color{#000000 }{ \displaystyle p(s)=-20 (s^2 - 70s + 600) }\)

is what you need at this point.

Answer 19

Okay i got that

Answer 20

you need to complete the square...

What would you add
\(\color{#000000 }{ \displaystyle s^2 - 70s + ~? }\)

to make a perfect square trinomial?

Answer 21

Note that,
\(\color{#000000 }{ \displaystyle x^2+2ax+a^2\color{grey}{~~~~~=(x+a)^2}}\)
is something that you want to obtain

Answer 22

so the 2a corresponds (in our case), to -70, right?

Answer 23

It's not a perfect square, right?

Answer 24

the x²+2ax+a² is a perf. sq.

Answer 25

your polynomial inside the parenthesis is not,,,

Answer 26

Yea, in the parenthesis, the first term and last term are perfect squares, but the middle one isn't, right?

Answer 27

but if you tell me the missing number for
\(\color{#000000 }{ \displaystyle s^2 - 70s +~{\rm what?} }\)

then I can show you a trick.

Answer 28

"what?" will make the s²-70s a perfect square trinomial when added?

Answer 29

I don't know

Answer 30

\(s^2+2as+a^2\)

is the form you want

Answer 31

you have
\(s^2-70s\)

Answer 32

So -70s, corresponds to the "2as" peace, right?

Answer 33

So, what is the "a" in our case?

Answer 34

-35?

Answer 35

yup

Answer 36

and "a²" is what?

Answer 37

YAY!

Answer 38

ok, next question i asked...

Answer 39

(-35)^2

Answer 40

yes, and that would be?

Answer 41

1225

Answer 42

yup

Answer 43

So, I will show you the trick now...

Answer 44

\(\color{#000000 }{ \displaystyle p(s)=-20 (s^2 - 70s + 600) }\)

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.

Answer 45

\(\color{#000000 }{ \displaystyle p(s)=-20 (s^2 - 70s + 600+625) +(-20)(-625)}\)
\(\color{#000000 }{ \displaystyle p(s)=-20 (s^2 - 70s + 1225) +1250}\)

and recall the form, \(s^2+2as+a^2=(s+a)^2\)

We have already clarified that:
a=-35
a²=1225
2a=-70

\(\color{#000000 }{ \displaystyle p(s)=-20 (s^2 - 2(35)s +(-35)^2) +1250}\)

Answer 46

And we apply the form now!

\(\color{#000000 }{ \displaystyle p(s)=-20 (s - 35)^2 +1250}\)

Answer 47

when you digest this info, let me know...

Answer 48

One moment, still digesting

Answer 49

I got it. What's next?

Answer 50

\(\color{#000000 }{ \displaystyle p(s)=-20 (s - 35)^2 +1250}\)

can you tell me if this parabola opens up or down (and why)?

Answer 51

down? i think..

Answer 52

yes, and why?

Answer 53

negative coefficient?

Answer 54

yes, the leading coefficient is negative...
fabulous!

Answer 55

Since the parabola opens down, the vertex is the maximum point of the parabola. (The other points are all lower)

Answer 56

So you need to find the vertex, and for this there is a rule...

Answer 57

\(\color{#000000 }{ \displaystyle y(x)=a(x-k)^2+h }\)
will have a vertex of \(\color{#000000 }{ \displaystyle (h,k) }\)

Answer 58

(1250,-35)?

Answer 59

Sorry (1250, 35)

Answer 60

not k,h

the other way around

Answer 61

my fault

Answer 62

\(\color{#000000 }{ \displaystyle y(x)=a(x-h)^2+k }\)
will have a vertex of \(\color{#000000 }{ \displaystyle (h,k) }\)

Answer 63

that is the form

Answer 64

(35, 1250)

Answer 65

yes, correct, and I apologize for my mistake.

Answer 66

No worries, you have been amazing help!

Answer 67

And guidance

Answer 68

And the vertex of (35,1225) indicates that the maximum profit is 1225, and it occurs when you sell 35 dollars per item.

Answer 69

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.

Answer 70

d?

Answer 71

you are close.

Answer 72

c?

Answer 73

can you sketch (somehow) a parabola that opens down please?

Answer 74

any picture that looks like opening-down parabola would suffice.