Max and Min Ques (1 Viewer)

[taco man]

Hello
can sum1 do this question for me
Suppose the cost of producing x items per hour is given by C(x) where
C(x)=x^2 + 10, and the number of items sold per hour at a price p dollars per item is x= 16-p:
a)Find in terms of x the revenue gained from the sales
b)Hence show that the profit achieved per hour is given by -2x^2+16x-10
c)Find the number of items that should be produced each hour in order to maximise the profit
d)Find the maximum profit
thx

 

[BlackJack]

Do this:
Set out your two equations,
C(x)=x^2 + 10
x= 16-p
Now, p is the selling price, C is the cost, x represents a number of items per hour. Your total gross profit every hour is selling price per item * items sold per hour. Which is x*p. You have assumed that you are selling those items as fast as youu're producing them.

Net profit (revenue I guess) per hour:
is gross profit - cost
= xp - C(x)
= xp - x² - 10

Rearrange x = 16 - p to get p = 16 - x, and substitute this p.

= x*(16 - x) - x² - 10
= 16x - 2x² - 10

Have I understood your question correctly?

 

[taco man]

hmmm.. ur answer looks correct and it should be, but in the answers it says x(16-x)

 

[Slidey]

Can you type out your question more clearly, ensuring you state all the information provided?

 

[taco man]

yea ill do it now, ill add parts b) c) and d)












Edit:I typed out the question as it exactly is in the textbook, if it helps its question 18, Exercise 8E from the Cambridge 3 unit Yr11 textbook

 

[Slidey]

Hello
can sum1 do this question for me
Suppose the cost of producing x items per hour is given by C(x) where
C(x)=x^2 + 10, and the number of items sold per hour at a price p dollars per item is x= 16-p:
a)Find in terms of x the revenue gained from the sales
b)Hence show that the profit achieved per hour is given by -2x^2+16x-10
c)Find the number of items that should be produced each hour in order to maximise the profit
d)Find the maximum profit
thx

C(x)=x^2+10
x=16-p --> p=16-x)

Cost=C(x)=x^2+10
Since you sell x items per hour at p dollars, you make xp dollars, so profit = xp
Nett gain = profit - cost = xp - x^2 + 10 = x(16-x) - x^2 - 10
Nett gain = -2x^2+16x-10. And I got the same answer as McLake. Thought I might.

Ah! I see. Our problem is that were assume it is asking for nett revenue (revenue after cost has been factored in). Not so. It simply wants the amount of money you make ignoring costs. This is equal to x*p, but p=16-x, so revenue = x(16-x).

For part b), see mine or McLake's working.

c) num items needed to max profit:
P(x) = -2x^2+16x-10
P'(x) = -4x+16
For turning points or maximum and min values, set the derivative to zero:
P'(x)=-4x+16=0
-4x=-16
x=4.

So the maximum profit is acheived when 4 items are sold per hour.

To find max profit, go P(4)=-2*(4)^2+16*4-10=-32+64-10=32-10=22
Max profit = 22

Alternatively, complete the square:
-2x^2+16x-10
-2(x^2-8x+5)
-2(x^2-8x+16-11)
-2(x-4)^2 + 22
.'. at when 4 items are sold, the maximum profit of 22 is reached.