Answered

Department stores are about to release a new makeup line, and a promotional kit will be sold for [tex]\$42.75[/tex]. The cost to produce the kits can be determined by the cost function, where [tex]x[/tex] is the number of kits to be produced and sold:
[tex]
C(x) = 28.25x + 8,250
[/tex]

Type the correct answer in the box.
To the nearest whole number, the company must sell [tex]\square[/tex] makeup kits before it starts to earn a profit.



Answer :

GinnyAnswer
To determine the number of makeup kits that must be sold before the company starts to earn a profit, we need to find the break-even point. The break-even point is where the total cost of production equals the total revenue from sales. Here are the steps:

1. Identify the cost function and revenue function:
- The cost function [tex]\( C(x) \)[/tex] is given by:
[tex]\[ C(x) = 28.25x + 8250 \][/tex]
where [tex]\( x \)[/tex] is the number of kits produced and sold.
- The revenue function [tex]\( R(x) \)[/tex] is:
[tex]\[ R(x) = 42.75x \][/tex]
where [tex]\( 42.75 \)[/tex] is the sale price per kit.

2. Set the total cost equal to the total revenue to find the break-even point:
We need to solve the equation [tex]\( C(x) = R(x) \)[/tex]:
[tex]\[ 28.25x + 8250 = 42.75x \][/tex]

3. Isolate [tex]\( x \)[/tex] by rearranging the equation:
[tex]\[ 8250 = 42.75x - 28.25x \][/tex]
[tex]\[ 8250 = 14.5x \][/tex]

4. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{8250}{14.5} \][/tex]
This gives:
[tex]\[ x \approx 568.9655172413793 \][/tex]

5. Since we need a whole number of kits, round up to the nearest whole number:
[tex]\[ x \approx 569 \][/tex]

Therefore, the company must sell 569 makeup kits before it starts to earn a profit.

Other Questions