Question

Randex Manufacturing estimates that its weekly profit, P, in hundreds of dollars, can be approximated by the formula P = –2x2 + 4x + 2 where x is the number of units produced per week, in thousands.


a. How many units should the company produce per week to earn the maximum profit?


b. Find the maximum weekly profit.

a. 10 units; b. $1000

a. 100 units; b. $800

a. 1000 units; b. $400

a. 1 unit; b. $500

Answer 1

@freckles

Answer 2

to earn max profit think about the max of the graph
where is the max point located on a parabola?
what is that famous v word?

Answer 3

vertex

Answer 4

yep the parabola is open down because of the negative coefficent on the x^2 part
so we definitely have a max
and we know max/min occur at the vertex for a parabola

Answer 5

do you know how to find the vertex

Answer 6

no

Answer 7

do you know how to complete the square?

Answer 8

\[P=-2x^2+4x+2 \\ P=-2(x^2-2x)+2 \\ P=-2(x^2-2x+?-?)+2 \\ P=-2(x^2-2x+?)+(-2)(-?)+2 \\ P=-2(x^2-2x+?)+2?+2\]
so x^2-2x+what number will give me a complete square

Answer 9

100

Answer 10

@freckles

Answer 11

is the answer b

Answer 12

how did you get a 100?

Answer 13

actually I guess instead of the choices being labeled a,a,a,a I should imagine it as a,b,c,d :p

Answer 14

yeah

Answer 15

and i am just guessing

Answer 16

so you aren't able to answer my one question
x^2-2x+what number is going to give me a complete square?

Answer 17

no how would i do that

Answer 18

\[x^2+kx+(\frac{k}{2})^2=(x+\frac{k}{2})^2\]

Answer 19

what is k

Answer 20

@freckles

Answer 21

you compare it to what I gave you to determine k so you can figure out what number to add in to complete the square

Answer 22

\[x^2-2x \text{ is the same as } x^2+kx \text{ when } k=?\]

Answer 23

2

Answer 24

\[x^2+kx+(\frac{k}{2})^2=(x+\frac{k}{2})^2 \\ \text{ well } k=-2 \text{ not} 2 \\ x^2+kx \text{ is the same as } x^2-2x \text{ when } k=-2 \\ \text{ not just use the formula to complete the \square } \\ x^2+-2x+(\frac{-2}{2})^2=(x+\frac{-2}{2})^2\]
you can simplify a bit

Answer 25

\[P=-2x^2+4x+2 \\ P=-2(x^2-2x)+2 \\ P=-2(x^2-2x+?-?)+2 \\ P=-2(x^2-2x+?)+(-2)(-?)+2 \\ P=-2(x^2-2x+?)+2?+2 \\ P=-2(x^2-2x+1)+2(1)+2 \\ P=-2(x-1)^2+2(1)+2\]

Answer 26

this is vertex form
\[P=a(x-h)^2+k \text{ where } (h,k) \text{ is vertex }\]

Answer 27

so its 1

Answer 28

(x,P)
where x represents number of units produced each week in thousands
and
P represents profit in hundreds

so when you say 1 what do you mean by 1?

Answer 29

yes the x-coordinate of the vertex is 1
what does that mean though in terms of our problem

Answer 30

1 unit for 500 ?

Answer 31

are you understanding the x-coordinates are in thousands
in the P-coordinate also known as the y-coordinate is in hundreds
also how did you get 500?

Answer 32

so the x-coordinate being 1 means you have how many units?

Answer 33

2(1)+2
is
2+2
which is 4

maybe you did 2+1+2 or something not sure....


but the x-coordinate being 1 means?
and the y-coordinate being 4 means?

Answer 34

pretend I got the x-coordinate was 2
then that would mean for this problem I have 2000 units is when I get the max profit
where the corresponding y value would be the max in hundreds

Answer 35

that is just an example

Answer 36

I'm just asking you to interpret our answer in terms of your problem

Answer 37

so 1 means 1000

Answer 38

the x-coordinate of the vertex being 1 means we need to produce 1000 units to achieve our max profit

Answer 39

and the y or P-coordinate which is 2(1)+2=4 is in hundreds (dollars)

Answer 40

so what is our max profit ?

Answer 41

so 1000 units 400$

Answer 42

yes
producing 1000 units will achieve our max profit of 400 dollars

Answer 43

@freckles

Answer 44

yes?

Answer 45

?

Answer 46

do you have another question?

Answer 47

if you posted another question I really do not see it...

Answer 48

no i'm all set thank you so much

Answer 49

ok I was confused