The revenue function [tex]R(x)[/tex] and the cost function [tex]C(x)[/tex] for a particular product are given. These functions are valid only for the specified range of values. Find the number of units that must be produced to break even.
[tex]R(x) = 200x - x^2; C(x) = 10x + 8125; 0 \leq x \leq 100[/tex]
Answer (2)
Set the revenue function equal to the cost function: 200 x − x 2 = 10 x + 8125 .
Rearrange the equation to form a quadratic equation: x 2 − 190 x + 8125 = 0 .
Solve the quadratic equation using the quadratic formula: x = 2 190 ± ( − 190 ) 2 − 4 ( 1 ) ( 8125 ) .
Determine the valid solution within the range 0 ≤ x ≤ 100 : 65 .
Explanation
Understanding the Problem We are given the revenue function R ( x ) = 200 x − x 2 and the cost function C ( x ) = 10 x + 8125 . We want to find the number of units x that must be produced to break even. This means we need to find the value(s) of x for which the revenue equals the cost, i.e., R ( x ) = C ( x ) . The functions are valid for 0 ≤ x ≤ 100 .
Setting up the Equation To break even, the revenue must equal the cost. So, we set R ( x ) = C ( x ) : 200 x − x 2 = 10 x + 8125
Forming the Quadratic Equation Rearrange the equation to form a quadratic equation: x 2 − 190 x + 8125 = 0
Applying the Quadratic Formula Now, we solve the quadratic equation for x using the quadratic formula: x = 2 a − b ± b 2 − 4 a c
where a = 1 , b = − 190 , and c = 8125 .
Calculating the Discriminant and Roots Plugging in the values, we get: x = 2 ( 1 ) 190 ± ( − 190 ) 2 − 4 ( 1 ) ( 8125 )
x = 2 190 ± 36100 − 32500
x = 2 190 ± 3600
x = 2 190 ± 60
Finding the Possible Solutions So, the two possible solutions for x are: x 1 = 2 190 + 60 = 2 250 = 125
x 2 = 2 190 − 60 = 2 130 = 65
Checking the Validity of Solutions We need to check if the solutions are within the valid range 0 ≤ x ≤ 100 . x 1 = 125 is not within the range, since 100"> 125 > 100 . x 2 = 65 is within the range, since 0 ≤ 65 ≤ 100 . Therefore, the number of units that must be produced to break even is 65.
Final Answer The number of units that must be produced to break even is 65.
Examples
Understanding break-even points is crucial in business. For example, a small bakery can use this concept to determine how many cakes they need to sell each month to cover their costs, including ingredients, rent, and utilities. By calculating the break-even point, the bakery owner can set realistic sales goals and make informed decisions about pricing and production levels. This ensures the bakery remains profitable and sustainable in the long run. The break-even point helps in planning and managing the business effectively.
To break even, set the revenue function equal to the cost function and solve for x . This results in a quadratic equation, yielding two solutions. The valid solution within the range is 65 units.
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