Answer :
To solve the problem, follow these steps:
1. First, write the revenue and cost functions. The revenue from selling [tex]$x$[/tex] pastries is given by
[tex]$$\text{Revenue} = 1.60x.$$[/tex]
The total cost is the sum of the fixed cost and the variable cost per pastry:
[tex]$$\text{Total Cost} = 720 + 0.70x.$$[/tex]
2. The profit function, which is the revenue minus the total cost, is:
[tex]$$P(x) = 1.60x - (720 + 0.70x).$$[/tex]
3. Simplify the profit function by combining like terms:
[tex]$$P(x) = 1.60x - 0.70x - 720 = 0.90x - 720.$$[/tex]
4. To find the break-even point, set the profit function equal to zero and solve for [tex]$x$[/tex]:
[tex]$$0.90x - 720 = 0.$$[/tex]
Add [tex]$720$[/tex] to both sides:
[tex]$$0.90x = 720.$$[/tex]
Divide both sides by [tex]$0.90$[/tex]:
[tex]$$x = \frac{720}{0.90} = 800.$$[/tex]
Thus, the final answers are:
a. The linear profit function is
[tex]$$P(x) = 0.90x - 720.$$[/tex]
b. The break-even point is at [tex]$800$[/tex] pastries.
1. First, write the revenue and cost functions. The revenue from selling [tex]$x$[/tex] pastries is given by
[tex]$$\text{Revenue} = 1.60x.$$[/tex]
The total cost is the sum of the fixed cost and the variable cost per pastry:
[tex]$$\text{Total Cost} = 720 + 0.70x.$$[/tex]
2. The profit function, which is the revenue minus the total cost, is:
[tex]$$P(x) = 1.60x - (720 + 0.70x).$$[/tex]
3. Simplify the profit function by combining like terms:
[tex]$$P(x) = 1.60x - 0.70x - 720 = 0.90x - 720.$$[/tex]
4. To find the break-even point, set the profit function equal to zero and solve for [tex]$x$[/tex]:
[tex]$$0.90x - 720 = 0.$$[/tex]
Add [tex]$720$[/tex] to both sides:
[tex]$$0.90x = 720.$$[/tex]
Divide both sides by [tex]$0.90$[/tex]:
[tex]$$x = \frac{720}{0.90} = 800.$$[/tex]
Thus, the final answers are:
a. The linear profit function is
[tex]$$P(x) = 0.90x - 720.$$[/tex]
b. The break-even point is at [tex]$800$[/tex] pastries.
