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A jewelry artisan has determined that her revenue, $y$, each day at a craft fair is at most $-0.5 x^2+30 x$, where $x$ represents the number of necklaces she sells during the day. To make a profit, her revenue must be greater than her costs, $2 x+150$.
Write a system of inequalities to represent the values of $x$ and $y$ where the artisan makes a profit. Then complete the statements.
The point $(30,230)$ is
Answer (1)
The system of inequalities is y − S u b s t i t u t e t h e p o in t (30, 230)$ into the first inequality: $230
Substitute the point ( 30 , 230 ) into the second inequality: 2(30) + 150"> 230 > 2 ( 30 ) + 150 , which simplifies to 210"> 230 > 210 , which is true.
Since both inequalities are true for the point ( 30 , 230 ) , the point is a solution to the system of inequalities.
Explanation
Understanding the Problem The artisan's revenue, denoted as y , is at most − 0.5 x 2 + 30 x , where x represents the number of necklaces sold. To make a profit, the revenue y must exceed the costs, which are given by 2 x + 150 . We need to formulate a system of inequalities that captures the conditions for the artisan to make a profit and then evaluate whether the point ( 30 , 230 ) satisfies this system.
Formulating the System of Inequalities The system of inequalities can be written as:
y ≤ − 0.5 x 2 + 30 x (Revenue constraint)
2x + 150"> y > 2 x + 150 (Profit condition)
Checking the First Inequality Now, let's substitute the point ( 30 , 230 ) into the first inequality:
230 ≤ − 0.5 ( 30 ) 2 + 30 ( 30 )
Simplifying the right side:
− 0.5 ( 900 ) + 900 = − 450 + 900 = 450
So, the first inequality becomes:
230 ≤ 450
This statement is true.
Checking the Second Inequality Next, substitute the point ( 30 , 230 ) into the second inequality:
2(30) + 150"> 230 > 2 ( 30 ) + 150
Simplifying the right side:
60 + 150 = 210
So, the second inequality becomes:
210"> 230 > 210
This statement is also true.
Conclusion Since both inequalities are satisfied by the point ( 30 , 230 ) , this point is a solution to the system of inequalities. Therefore, at x = 30 necklaces sold, with a revenue of y = 230 , the artisan makes a profit.
Examples
Consider a scenario where a vendor at a local market sells handmade scarves. The vendor's revenue depends on the number of scarves sold, and they need to ensure their revenue exceeds their costs to make a profit. By setting up a system of inequalities similar to the artisan's situation, the vendor can determine the range of scarf sales needed to achieve profitability. This approach helps in making informed decisions about pricing, production, and sales targets to ensure the vendor's business remains sustainable and profitable.
