a custom printing store is planning on adding painters caps to its product line. For the first year the fixed costs for setting up production are $15,000 the variable costs for producing a dozen caps are $8 the revenue on each dozen caps will be $20 find the total profit p(x)=

the break even point is

10

To find the total profit, we need to consider the total revenue and total cost.

Total Cost = Fixed Costs + (Variable Costs per unit * Quantity produced)
Total Cost = $15,000 + ($8 * x)

Total Revenue = Price per unit * Quantity sold
Total Revenue = $20 * x

Total profit, P(x), can be calculated by subtracting the total cost from the total revenue.
P(x) = Total Revenue - Total Cost
P(x) = ($20 * x) - ($15,000 + ($8 * x))
P(x) = $20x - $15,000 - $8x
P(x) = $12x - $15,000

To find the break-even point, we set the profit equal to zero and solve for x.
0 = $12x - $15,000
$12x = $15,000
x = $15,000 / $12
x = 1250 dozen caps

Therefore, the break-even point is 1250 dozen caps.

To find the total profit, we need to consider both the fixed costs and the variable costs.

Total Cost (TC) can be determined using the formula: TC = Fixed Costs + (Variable Cost per unit x Quantity)

In this case, the fixed costs are $15,000, and the variable cost is $8 per dozen caps. Let's assume the quantity of caps produced is represented by 'x'.

So, TC = $15,000 + ($8 x x)

Next, we need to calculate the Total Revenue (TR). The revenue per dozen caps is $20. Since one dozen equals 12 caps, the revenue per cap is $20/12.

TR = ($20/12) x x

Finally, we can calculate the Profit (P) using the formula: P = TR - TC.

P(x) = TR - TC = ($20/12) x x - [$15,000 + ($8 x x)]

Now, let's simplify the equation to find the total profit.

P(x) = ($20/12) x - $15,000 - $8x
P(x) = ($5/3) x - $15,000 - $8x

To find the break-even point, we need to determine the quantity at which the profit becomes zero. In other words, we need to find the value of 'x' that makes P(x) = 0.

($5/3) x - $15,000 - $8x = 0

Simplifying the equation further would allow us to solve for 'x' and determine the break-even point.