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Answer :
To find how many umbrellas the company must sell to make more than [tex]$20,000, we can set up an inequality based on their profit.
1. Define the terms:
- Let \( x \) represent the number of umbrellas sold.
- The company makes $[/tex]75 per umbrella, so the revenue from selling [tex]\( x \)[/tex] umbrellas is [tex]\( 75x \)[/tex].
- The cost to design and produce the umbrellas is [tex]$16,000.
2. Set up the inequality for the profit:
- Profit is calculated as the revenue minus the cost, which needs to be greater than $[/tex]20,000.
- The inequality is: [tex]\( 75x - 16000 > 20000 \)[/tex].
3. Solve the inequality:
Start with the inequality:
[tex]\[
75x - 16000 > 20000
\][/tex]
Add [tex]$16,000 to both sides to isolate the term with \( x \):
\[
75x > 20000 + 16000
\]
Simplify the right side:
\[
75x > 36000
\]
Finally, divide both sides by 75 to solve for \( x \):
\[
x > \frac{36000}{75}
\]
Calculate the division:
\[
x > 480
\]
Therefore, the company must sell more than 480 umbrellas to achieve a profit greater than $[/tex]20,000.
The correct inequality is represented by option (A): [tex]\( 75x - 16000 > 20000 \)[/tex].
1. Define the terms:
- Let \( x \) represent the number of umbrellas sold.
- The company makes $[/tex]75 per umbrella, so the revenue from selling [tex]\( x \)[/tex] umbrellas is [tex]\( 75x \)[/tex].
- The cost to design and produce the umbrellas is [tex]$16,000.
2. Set up the inequality for the profit:
- Profit is calculated as the revenue minus the cost, which needs to be greater than $[/tex]20,000.
- The inequality is: [tex]\( 75x - 16000 > 20000 \)[/tex].
3. Solve the inequality:
Start with the inequality:
[tex]\[
75x - 16000 > 20000
\][/tex]
Add [tex]$16,000 to both sides to isolate the term with \( x \):
\[
75x > 20000 + 16000
\]
Simplify the right side:
\[
75x > 36000
\]
Finally, divide both sides by 75 to solve for \( x \):
\[
x > \frac{36000}{75}
\]
Calculate the division:
\[
x > 480
\]
Therefore, the company must sell more than 480 umbrellas to achieve a profit greater than $[/tex]20,000.
The correct inequality is represented by option (A): [tex]\( 75x - 16000 > 20000 \)[/tex].
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