maximizing profits: R(x) = -0.5x^2 + 50x, C(x) = 4x + 10

[Tazman]

Can someone lend me a hand?

Revenue function - R(x) = -0.5x^2 + 50x
Cost function - C(x) = 4x+10

A. What would the profit be for producing and selling 10 scarves?

B. What would the maximum profit be?

C. How many scarves would need to be sold to maximize the profits?

Any step by step instructions would help me out alot. Its a textbook question.

thx

 

[o_O]

Here I'll give you some hints (assuming that my assumptions are right) and what you can figure out for yourself first. Unless someone else decides they want to answer all the questions for you =/

A. Profits = Revenue - Cost. What's the revenue from selling 10 scarves? How about producing them?

B. The graph of the "profits" function is going to be a parabola opening DOWNWARDS (meaning it's going to have a maximum value) since it's going to be in the form of ax² + bx + c where a = -.5. From what you know from calculus, how can you find this maximum?

C. Use the information from B. For me, I'd probably wind up finding the answer to this question first.

 

[Tazman]

Here I'll give you some hints (assuming that my assumptions are right) and what you can figure out for yourself first. Unless someone else decides they want to answer all the questions for you =/

A. Profits = Revenue - Cost. What's the revenue from selling 10 scarves? How about producing them?

B. The graph of the "profits" function is going to be a parabola opening DOWNWARDS (meaning it's going to have a maximum value) since it's going to be in the form of ax² + bx + c where a = -.5. From what you know from calculus, how can you find this maximum?

C. Use the information from B. For me, I'd probably wind up finding the answer to this question first.

Answer to C. So I would find the Answer to C First which is:

50x-.5x^2- (4x+10)
-.5x^2+46x-10
Then I would take the derivative and set it to zero: -1x+46=0
Number of Units to be sold to maximize is: 46 units?


Answer to B: What would the maximum profit be?
F(x)= 46
Maximum Profit would be: $1048?

Answer to A: What would the profit be for producing and selling 10 scarves:
P(x) = -.5(10)^2 + 50(10) - (4(10) +10)

Profit for selling and producing 10 scarves: $500

I know I took this step by step - but getting familar with calc.

Can you let me know how I did.
Thx

 

[o_O]

C. Yep.
B. Just a little detail on your notation. It's not f(x) = 46. It's f(46) = 1048
A. Well you didn't have to do this last because this isn't really relevant to the other questions but doesn't matter : ) One thing: Maybe you plugged in something wrong (or I did) but I got f(10) = $400

Nice work!

 

[Tazman]

C. Yep.
B. Just a little detail on your notation. It's not f(x) = 46. It's f(46) = 1048
A. Well you didn't have to do this last because this isn't really relevant to the other questions but doesn't matter : ) One thing: Maybe you plugged in something wrong (or I did) but I got f(10) = $400

Nice work!

Thx for the help.

On the answer for Question A - If I am given the R(x) and C(x) functions, Im really only replacing the x values with the number, such as 10 scarves in this case. That seems easy enough.

dont mind me - again, just getting use to this new thing..

thx

 

[o_O]

Np. For A, you said you got $500 but I think I'm sure it's $400:
P(10) = -.5(10)² + 50(10) - [4(10) + 10]
P(10) = -50 + 500 - (50)
P(10) = 500 - 100
P(10) = 400

 

[Tazman]

Np. For A, you said you got $500 but I think I'm sure it's $400:
P(10) = -.5(10)² + 50(10) - [4(10) + 10]
P(10) = -50 + 500 - (50)
P(10) = 500 - 100
P(10) = 400

Your rights. Thx.. but I have the concept - gotta brush up with the normal adding and subtracting... lol