A printing company determines that to print a large format poster, it costs $75 to set up the production process. Each poster that is printed costs $22 in material and labor. The printing company sells these posters to marketing firms for $60 each. Find a linear model P that models the profit P(x) from selling x posters. [Note: profit = revenue – cost]
P = (60 - 22) x - 75
= 38 x = 75
Well, let me put on my thinking clown hat for this one.
To find the profit from selling x posters, we need to subtract the cost from the revenue. The cost consists of the setup cost and the cost of printing each poster. So the cost function would be:
Cost(x) = 75 + 22x
The revenue is the number of posters sold multiplied by the selling price. So the revenue function is simply:
Revenue(x) = 60x
Now, to find the profit function, we subtract the cost function from the revenue function:
Profit(x) = Revenue(x) - Cost(x) = 60x - (75 + 22x)
Simplifying this, we get:
Profit(x) = 60x - 75 - 22x
And further simplifying:
Profit(x) = 38x - 75
So, the linear model that represents the profit P(x) from selling x posters is:
P(x) = 38x - 75
And that's how the clown calculates it!
To find the linear model, let's break down the cost and revenue for each poster.
Cost per poster: Set-up cost + Material and Labor cost
Cost per poster = $75 + $22 = $97
Revenue per poster: Selling price per poster
Revenue per poster = $60
Profit per poster: Revenue per poster - Cost per poster
Profit per poster = $60 - $97 = -$37
So, the profit P(x) from selling x posters can be modeled using the linear equation:
P(x) = (Revenue per poster - Cost per poster) * x
P(x) = (-$37) * x
Therefore, the linear model that represents the profit P(x) from selling x posters is:
P(x) = -37x
To find the linear model that models the profit from selling x posters, we need to subtract the cost from the revenue.
The revenue from selling x posters is given by the selling price multiplied by the number of posters sold. In this case, the selling price is $60 per poster, so the revenue is 60x.
The cost includes the setup cost and the cost per poster. The setup cost is a fixed cost that does not depend on the number of posters sold, which is $75. The cost per poster is $22, so the cost is 75 + 22x.
Therefore, the profit P(x) is given by:
P(x) = revenue - cost = 60x - (75 + 22x).
Simplifying the expression:
P(x) = 60x - 75 - 22x.
Combining like terms:
P(x) = 38x - 75.
So the linear model P that models the profit from selling x posters is P(x) = 38x - 75.