SOLUTION: Suppose the demand price for selling out a production run of x DVDs is given by p(x)=-0.0005x^2+60 dollars per DVD. Further suppose the weekly cost of producing these DVDs is given

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Question 1195270: Suppose the demand price for selling out a production run of x DVDs is given by p(x)=-0.0005x^2+60 dollars per DVD. Further suppose the weekly cost of producing these DVDs is given by C(x)=-0.001x^2+18x+4000 dollars to produce that many DVDs.

(i) Write the Revenue function R(x). And be sure to state its DOMAIN.
(ii) Use (i) to write the profit function P(x). Again, be sure to state its DOMAIN.

I am very confused as to how to get the domain in these problems because initially I thought it was (-infinity, infinity) for both revenue function and profit function, but my professor said that mathematically this is correct, but since we're talking about the real world it would be something else and it wouldn't make sense to have negative infinity, and I would like an explanation as to how to get the domain. Would the interval be (0, infinity) for both, or is there a method to figuring out the domain in this situation.

Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

the demand price for selling out a production run of x DVDs is given by
p%28x%29=-0.0005x%5E2%2B60

the weekly cost of producing is given by
C%28x%29=-0.001x%5E2%2B18x%2B4000+
so, we have
demand price: +p%28x%29=-0.0005x%5E2%2B60+
weekly cost : C%28x%29=-0.001x%5E2%2B18x%2B4000

i)
the Revenue function R%28x%29:
Revenue is equal to the number of units sold times the price per unit. To obtain the revenue function, multiply the output level by the price function.

R%28x%29=+x%2Ap%28x%29
R%28x%29=+x%2A%28-0.0005x%5E2%2B60%29
R%28x%29=+-0.0005x%5E3%2B60x

DOMAIN: R+(all real numbers)

so, domain is all x+%3E=0
interval notation:
[0, infinity)

ii)

The profit a business makes is equal to the revenue it takes in minus what it spends as costs. To obtain the profit function, subtract costs from revenue.

P%28x%29=R%28x%29+-+C%28x%29
P%28x%29=-0.0005x%5E3%2B60x+-+%28-0.001x%5E2%2B18x%2B4000%29
P%28x%29=-0.0005x%5E3%2B60x+%2B0.001x%5E2-18x-4000
P%28x%29=-0.0005x%5E3+%2B+0.001x%5E2+%2B+42x+-+4000

DOMAIN: R (all real numbers)

so, domain is all x+%3E=0
interval notation:
[0, infinity)

Answer by ikleyn(52824) About Me  (Show Source):
You can put this solution on YOUR website!
.

Hello,

formally, you correctly determined the domains of your functions as polynomials.

But since your professor refers to common sense and real life, I think that you should
to NARROW the domains to the areas where x is non-negative (x >= 0), the price function is positive
and where the cost function is positive.


These are not formal instructions, but those that are dictated by COMMON SENSE.


Think about it . . .