Question

Find the price per unit p that produces the maximum profit P.
C = 10x + 400 (Cost Function)
p = 16 – 0.1√x (Demand function)
$ ?

Answer 1

So far I got worked out: P=R-C so,
xp-(10x+400)
=x(16-0.1sqrtx)-10x+400
=16-0.1sqrtx-10x+400? Am I doing this right?
What do I do now?

Answer 2

I really need help :(

Answer 3

what is R? x*p?

Answer 4

now you have profit as a function of x, maximize using calculus techniques (if you know them not sure)

Answer 5

I believe he is saying R is revenue, that is quantity demanded times price.

Answer 6

anyway what you are doing is wrong, it should be "xp-10x-400" as the minus is multiplying the whole term "10x-400"

Answer 7

^^whoops she is right, I didn't catch that :P, but yeah take the function and maximize it, I'm not sure what level math you have had, or which econ this is though but thats how you can get the correct answer

Answer 8

A quick edit, you will have found the maximum demand for the product, so you need to plug it back in to the demand function to get the price.

Answer 9

I'm not sure how I'm doing it wrong.. i do have x*p p=16-0.1sqrtx... but I don't know what to do now..

Answer 10

I don't know how to maximize :/

Answer 11

This question is from my calculus homework. I'm taking it online and I have no help besides this study place, I need help walking me though the whole thing

Answer 12

I have x(16-0.1sqrtx)-10x+400. what do I do with this?

Answer 13

its x(16-0.1sqrtx)-10x-400

First multiply that first x out. It helps if you write it like such
\[\frac{ x ^{1/2}}{ 10 }\]
before multiplying out (:P you can add the exponents more easily)

Answer 14

Next take d(Profit)/dx

remember x is the quantity.

Set it equal to zero. Because this is an econ question its safe to say this value will be the maximum, but usually you'd need to check it using a test (first derivative or second derivative).

Answer 15

finally plug in your result into the demand function to find the p.

Answer 16

so now it'S x^1/2/10=0?

Answer 17

oh sorry I must have mislead you. I just told you write .1sqrtx in a different way to help you. you still need to multiply the whole thing out. go ahead and do it, tell me if you need help.

Answer 18

ok, so i have 16-(x^1/2/10)-10x-400=0.. so x must equal..

Answer 19

at least 0 right?

Answer 20

nono hold it

Answer 21

first take the expression and multiply it out, you get 16x- (1/10)x^3/2-10x-400

Answer 22

where does the 1 come from?

Answer 23

the 1/10x^3/2?

Answer 24

uh im just writing x^3/2 divided by 10, its the same thing as .1x^3/2

Answer 25

http://i.imgur.com/axwjYnZ.png

Answer 26

ok.. so i start with 16x-(1/10)x^3/2-10x-400

Answer 27

This is a plot of the profit curve with respect to quantity demanded, you can see the functions above it
you are trying to find the max of this curve

Answer 28

good, now take your equation you just typed, and take the derivative with respect to x and set it to 0, aka find the critical points (in this case one point because I know it is one)

Answer 29

wow, I haven't even taken econ yet but I for sure can tell I will be confused when i take it lol

Answer 30

don't worry too much about econ, lol the first year undergrad classes are extremely math easy. I'm in intermediate econ, and that is FINALLY when we use calc to maximize functions. If you are in High school, your econ is far easier than this problem

Answer 31

if you are in college, like i said the first level classes are easy

Answer 32

i'm a junior in college lol.. I still have a couple bac core classes before I can even start what i want to.. like this calculus, econ and physics.. anyhow, question at hand..

Answer 33

i dunno how to take derivative of the first two terms because of the radical weird exponent stuff..

Answer 34

i know the 400 drops off and the -10x is just -10

Answer 35

good good, the first two terms you can take the derivative by using power rule. That is

D[x^n]=n*x^(n-1)

Answer 36

\[D[x^n]=n*x^{n-1}\]

Answer 37

what do fractions look like as exponents come down in front from power rule :/

Answer 38

the 16x also becomes 16, haha but i bet you knew that from what you said about the -10

Answer 39

yeah.. the 16 is the easy one too lol

Answer 40

fractions are the exact same, just follow the rule exactly, subtract one. 3/2-2/2=1/2

Answer 41

so it just drops down to as 1/2x?

Answer 42

so f' would be 16(1/2x)-10?

Answer 43

not exactly, the 16x was a separate term (look above it was added) and the rule states you multiply by the original exponent, then drop it down

Answer 44

so it would be \[6-\frac{ 3 }{ 2 }\frac{ p^{1/2} }{ 10 }\]

Answer 45

wow, that just looks so complicated..

Answer 46

^^ 16-10 simplifies to 6, and the 3/2 drops down while you subtract 1 from the exponent, as per the rule

Answer 47

you can simplify further ofc, by multiplying the fraction out.

Answer 48

\[6-\frac{ 3p^{1/2} }{ 20}\]

Answer 49

set this equal to zero and solve for p. OH i just noticed this, i accidentally switched p and x, sorry about that, it is still x

Answer 50

I'm bad at fractions and exponent.. everything else is fine for me..

Answer 51

ok.. so now i set this to 0 and multiply it out?

Answer 52

yeah, solve for x, (the p is x i accidentally switched it)

Answer 53

after you solve for x, plug it into the demand function to find p.

Answer 54

lol I actually used mathematica to model the problem and here is the answer for you to check. You should solve x=1600 and then plug it in and get the price to be 12 (as seen in the image)/
http://i.imgur.com/02b40VG.png

Answer 55

right, gotcha.. geez, i was way off, i only got to 90 so far.. i was like geez... i need a big number to get this to equal 0

Answer 56

do you need me to review the steps to solving it? I can do that for you too

Answer 57

i was just trying to figure out the x.. so i just plug in the 1600 to the original function and it gives me 12 right?

Answer 58

\[6=3\sqrt{x}/20\]
\[120=3\sqrt{x}\]
\[40=\sqrt{x}\]
x=1600

Answer 59

that last step is squaring both sides.

Answer 60

oooooh.. so thats the easiest way to find x? square both sides?

Answer 61

haha theres no surefire easy way, you just simplify the equation like you did in algebra. but in this case, you square both sides because the square root of x squared is x. so you need to square both sides to find x.

Answer 62

right lol.. i hope to get the hang of this.. Thank you so very much for explaining this to me.. i would not have figured this out without you!

Answer 63

just to review the complete answer is shown in the image. The quantity sold is 1600 the price is $12 and the profit is 2800.

Answer 64

awesome! Thank you!!! I checked out the image :)