To determine how many widgets Wind Up Corporation must sell to reach the breakeven point, we need to find the value of [tex]\( q \)[/tex] such that the expenses equal the revenue. Let's go through the steps for finding this breakeven point:
1. Write the expense equation:
The monthly expense equation is given by:
[tex]\[
E = 3.20q + 56,000
\][/tex]
where [tex]\( E \)[/tex] is the total expense and [tex]\( q \)[/tex] is the number of widgets produced and sold.
2. Write the revenue equation:
The revenue from selling the widgets is given by:
[tex]\[
R = 6.00q
\][/tex]
where [tex]\( R \)[/tex] is the total revenue and [tex]\( q \)[/tex] is the number of widgets sold.
3. Set the expense equal to the revenue:
To find the breakeven point, we set the expense [tex]\( E \)[/tex] equal to the revenue [tex]\( R \)[/tex]:
[tex]\[
3.20q + 56,000 = 6.00q
\][/tex]
4. Solve for [tex]\( q \)[/tex]:
To isolate [tex]\( q \)[/tex], follow these steps:
- Subtract [tex]\( 3.20q \)[/tex] from both sides to move all terms involving [tex]\( q \)[/tex] to one side of the equation:
[tex]\[
56,000 = 6.00q - 3.20q
\][/tex]
- Combine like terms on the right side:
[tex]\[
56,000 = 2.80q
\][/tex]
- Divide both sides of the equation by 2.80 to solve for [tex]\( q \)[/tex]:
[tex]\[
q = \frac{56,000}{2.80}
\][/tex]
5. Calculate [tex]\( q \)[/tex]:
[tex]\[
q = 20,000
\][/tex]
Therefore, Wind Up Corporation must sell 20,000 widgets to reach the breakeven point.
