The average cost to produce $x$ widgets from the Widget Factory is: [tex]C(x)=0.5 x^2-4 x+5008[/tex]
How many widgets can the company produce to keep costs at or below $10,000?
Answer (2)
Set up the inequality 0.5 x 2 − 4 x + 5008 ≤ 10000 .
Rearrange the inequality to x 2 − 8 x − 9984 ≤ 0 .
Find the roots of the quadratic equation x 2 − 8 x − 9984 = 0 using the quadratic formula, which are x = − 96 and x = 104 .
Since the number of widgets must be non-negative, the solution is 0 ≤ x ≤ 104 . Therefore, the company can produce at most 104 widgets.
Explanation
Problem Setup We are given the average cost function C ( x ) = 0.5 x 2 − 4 x + 5008 and we want to find the number of widgets x that can be produced such that the cost is at or below 10 , 000. T hi s m e an s w e n ee d t oso l v e t h e in e q u a l i t y C(x) \le 10000$.
Setting up the Inequality First, let's set up the inequality: 0.5 x 2 − 4 x + 5008 ≤ 10000
Rearranging the Inequality Now, let's rearrange the inequality: 0.5 x 2 − 4 x + 5008 − 10000 ≤ 0 0.5 x 2 − 4 x − 4992 ≤ 0
Simplifying the Inequality To simplify, multiply the inequality by 2: x 2 − 8 x − 9984 ≤ 0
Finding the Roots Now, we need to find the roots of the quadratic equation x 2 − 8 x − 9984 = 0 . We can use the quadratic formula: x = 2 a − b ± b 2 − 4 a c where a = 1 , b = − 8 , and c = − 9984 .
Applying the Quadratic Formula Plugging in the values, we get: x = 2 ( 1 ) 8 ± ( − 8 ) 2 − 4 ( 1 ) ( − 9984 ) x = 2 8 ± 64 + 39936 x = 2 8 ± 40000 x = 2 8 ± 200
Calculating the Roots So the roots are: x 1 = 2 8 − 200 = 2 − 192 = − 96 x 2 = 2 8 + 200 = 2 208 = 104
Determining the Feasible Region Since x represents the number of widgets, it must be non-negative. The inequality x 2 − 8 x − 9984 ≤ 0 holds between the roots, so we have − 96 ≤ x ≤ 104 . Since x must be non-negative, we have 0 ≤ x ≤ 104 . Therefore, the company can produce up to 104 widgets to keep costs at or below $10,000.
Final Answer The company can produce a maximum of 104 widgets to keep the costs at or below $10,000.
Examples
Understanding quadratic inequalities is crucial in business for optimizing production costs. For instance, a bakery can use this concept to determine how many cakes to bake to maximize profit while keeping production costs within a certain budget. By modeling costs and revenue as quadratic functions, the bakery owner can solve inequalities to find the optimal number of cakes. Similarly, a farmer can use quadratic inequalities to determine the amount of fertilizer to use to maximize crop yield without exceeding a certain cost. These applications highlight the practical importance of quadratic inequalities in making informed business decisions.
The company can produce a maximum of 104 widgets to keep costs at or below $10,000. This was determined by solving the inequality derived from the average cost function. The relevant quadratic inequality has roots at -96 and 104, leading to valid widget production values between 0 and 104.
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