A print shop makes bumper stickers for election campaigns. If x stickers are ordered (where x < 10,000), then the price per sticker is 0.28 − 0.000002x dollars, and the total cost of producing the order is 0.087x − 0.0000005x2 dollars.
Use the fact that
profit = revenue − cost
to express P(x), the profit on an order of x stickers, as a difference of two functions of x.
p(x)=?
well, you surely know that revenue=price*quantity. That means
R(x) = x(0.28-0.000002x)
C(x) = 0.087x-0.0000005x^2
P(x) = R(x)-C(x)
Well, let me put on my humor hat and give it a shot!
Alright, so we know that profit is equal to revenue minus cost. Let's break it down:
Revenue function: R(x) = x * price per sticker
Cost function: C(x) = total cost of producing the order
Now, let's plug these into our profit equation:
P(x) = R(x) - C(x)
But wait, we need to express P(x) as a difference of two functions of x. So let's simplify further:
P(x) = R(x) - C(x)
P(x) = (x * price per sticker) - (total cost of producing the order)
Now, let's substitute the given functions for price per sticker and total cost of producing the order:
P(x) = (x * (0.28 - 0.000002x)) - (0.087x - 0.0000005x^2)
And there you have it, P(x) expressed as a difference of two functions of x. Now go out there and make some bumper sticker profits!
To express P(x), the profit on an order of x stickers, as a difference of two functions of x, we need to subtract the cost function from the revenue function.
Revenue Function:
The revenue can be calculated by multiplying the price per sticker by the number of stickers ordered, which is given by:
Revenue = Price per sticker * Number of stickers
Revenue = (0.28 - 0.000002x) * x
Revenue = 0.28x - 0.000002x^2
Cost Function:
The cost of producing the order is given by:
Cost = 0.087x - 0.0000005x^2
Profit Function:
Profit = Revenue - Cost
P(x) = (0.28x - 0.000002x^2) - (0.087x - 0.0000005x^2)
Simplifying the expression:
P(x) = 0.28x - 0.000002x^2 - 0.087x + 0.0000005x^2
P(x) = (0.28x - 0.087x) + (-0.000002x^2 + 0.0000005x^2)
P(x) = 0.193x - 0.0000015x^2
Therefore, the profit function P(x) can be expressed as the difference of two functions of x:
P(x) = 0.193x - 0.0000015x^2.
To express the profit on an order of x stickers as a difference of two functions of x, we need to subtract the cost function from the revenue function.
Given:
Revenue = price per sticker * number of stickers
Cost = total cost of producing the order
Let's start by expressing the revenue function:
Revenue = (price per sticker) * (number of stickers)
Revenue = (0.28 - 0.000002x) * x
Revenue = 0.28x - 0.000002x^2
Now, let's express the cost function:
Cost = total cost of producing the order
Cost = 0.087x - 0.0000005x^2
Finally, we can express the profit function as the difference between the revenue and cost functions:
Profit = Revenue - Cost
Profit = (0.28x - 0.000002x^2) - (0.087x - 0.0000005x^2)
Profit = 0.193x - 0.0000015x^2
So, the profit function on an order of x stickers is given by p(x) = 0.193x - 0.0000015x^2.