Pat owns a factory that manufactures key chains. His weekly profit (in hundreds of dollars) is given by P(x)= -2x^2+60x-120, where x is the number of key chains sold.

a) what is the largest number of cases he can sell and still make a profit?

b) how many cases should he make in sell in order to maximize his profits?

Question

Pat owns a factory that manufactures key chains. His weekly profit (in hundreds of dollars) is given by P(x)= -2x^2+60x-120, where x is the number of key chains sold.

a) what is the largest number of cases he can sell and still make a profit?

b) how many cases should he make in sell in order to maximize his profits?

danjs · Answer

how many in a case of em

danjs · Answer

question / answer mismatch thing

hartnn · Answer

for a) we need to have a positive profit, 

so can you solve 
$ -2x^2+60x-120 > 0 $

that will give the largest number of "keys".
Once we know how many keys are there in a case, 
we can find the largest number of cases. 

Please provide all the required information :)

anonymous · Answer

This is why I am confused on this problem..that is the whole problem word for word...The answer for A is 27 and B is 15 I just don't understand how they get B.

hartnn · Answer

ok, that means we assume there is one key in one case!

hartnn · Answer

start by solving 
$-2x^2+60x-120 > 0$

need a helping hand with that?

anonymous · Answer

$$-60+/-\sqrt{(60^2-4(-2)(-120)} \over 2(-2)$$

hartnn · Answer

ohh, just saw you need help with only b) ?

to maximize the profit, we need to find the extrema of P(x).

that we do by finding the derivative and setting it to 0.

P'(x) = 0 
$\dfrac{d}{dx}[−2x^2+60x−120] = 0  $

hartnn · Answer

the easier way to approach $−2x^2+60x−120 > 0$
is to factor out -2 first! 
$-2 (x^2 - 30 x + 60) > 0$

now using the quadratic formula for $x^2 -30x +60$ would be easier, right? :)

anonymous · Answer

yep, my bad!

hartnn · Answer

go ahead with the solution, 
let me know if you get stuck...have you already got 27 for a) ?

anonymous · Answer

yes I did, just not sure what the derivative is.

anonymous · Answer

or is 27 the derivative

hartnn · Answer

ohh... you've not been taught derivatives yet?

have you been taught how to find the vertex of $ax^2+bx+c$ ??

anonymous · Answer

yes -b/2*a

hartnn · Answer

excellent!

so the x co-ordinate of vertex of x^2-30x+60 will give you the max profit!

anonymous · Answer

thank you so much!

hartnn · Answer

welcome ^_^

astrophysics · Answer

First thing that I thought of was the vertex haha