A company plans to sell bicycle helmets for $26 each. The company's business manager estimates that the cost, y, of making x helmets is a quadratic function with a y-intercept of 8,400 and a vertex of (500, 15,900).
x = number of helmets
y = amount in dollars
Which system models this situation?
A. y = 26x and y = 8,400(x-500)^2 + 15,900
B. y = 26x and y = -0.030(x-500)^2 + 15,900
C. y = x/26 and y = -0.030(x-500)^2 + 15,900
D. y = x/26 and y = 8,400(x-500)^2 + 15,900
How many helmets must the company make and sell to break even?
Answer (2)
The revenue is modeled by y = 26 x .
The cost is modeled by y = − 0.03 ( x − 500 ) 2 + 15900 .
Set revenue equal to cost: 26 x = − 0.03 ( x − 500 ) 2 + 15900 .
Solve for x to find the break-even point: 600 helmets.
Explanation
Understanding the Problem Let's analyze the problem. We are given that the company sells helmets for $26 each. The cost of making x helmets is a quadratic function with a y -intercept of 8400 and a vertex of ( 500 , 15900 ) . We need to find the system of equations that models this situation and the number of helmets the company must make and sell to break even.
Finding the Revenue Equation The revenue equation is given by the price per helmet times the number of helmets sold. Therefore, the revenue equation is y = 26 x .
Finding the Cost Equation The cost equation is a quadratic function with a vertex at ( 500 , 15900 ) . The vertex form of a quadratic equation is y = a ( x − h ) 2 + k , where ( h , k ) is the vertex. In this case, h = 500 and k = 15900 . So, the cost equation is y = a ( x − 500 ) 2 + 15900 .
Solving for a We are given that the y -intercept is 8400 . This means that when x = 0 , y = 8400 . We can use this information to find the value of a . Substituting x = 0 and y = 8400 into the cost equation, we get 8400 = a ( 0 − 500 ) 2 + 15900 . Simplifying, we have 8400 = 250000 a + 15900 . Solving for a , we get a = 250000 8400 − 15900 = 250000 − 7500 = − 0.03 .
The System of Equations So the cost equation is y = − 0.03 ( x − 500 ) 2 + 15900 . Therefore, the system of equations that models this situation is y = 26 x and y = − 0.03 ( x − 500 ) 2 + 15900 .
Finding the Break-Even Point To find the break-even point, we need to find the number of helmets x for which the revenue equals the cost. So, we set the revenue equation equal to the cost equation: 26 x = − 0.03 ( x − 500 ) 2 + 15900 .
Simplifying the Equation Expanding and simplifying the equation, we get 26 x = − 0.03 ( x 2 − 1000 x + 250000 ) + 15900 , which becomes 26 x = − 0.03 x 2 + 30 x − 7500 + 15900 . Rearranging the equation to form a quadratic equation, we get 0.03 x 2 − 4 x + 8400 = 0 . Multiplying by 100 to get rid of the decimal, we get 3 x 2 − 400 x − 840000 = 0 .
Solving the Quadratic Equation Now we solve the quadratic equation 3 x 2 − 400 x − 840000 = 0 for x using the quadratic formula: x = 2 a − b ± b 2 − 4 a c = 2 ( 3 ) 400 ± ( − 400 ) 2 − 4 ( 3 ) ( − 840000 ) = 6 400 ± 160000 + 10080000 = 6 400 ± 10240000 = 6 400 ± 3200 . The two possible solutions for x are x = 6 400 + 3200 = 6 3600 = 600 and x = 6 400 − 3200 = 6 − 2800 = − 466.67 . Since the number of helmets must be positive, we take the positive solution, x = 600 .
Final Answer Therefore, the company must make and sell 600 helmets to break even.
Examples
Understanding break-even points is crucial in business. For example, if you're starting a lemonade stand, you need to know how many cups you must sell to cover your costs (lemons, sugar, cups). Similarly, a car manufacturer needs to calculate how many cars they must sell to cover production costs like materials, labor, and marketing. By finding the break-even point, businesses can make informed decisions about pricing, production levels, and overall financial strategy. This concept applies to various scenarios, from small startups to large corporations, ensuring they operate sustainably and profitably.
The system that models the situation is: Revenue: y = 26 x and Cost: y = − 0.03 ( x − 500 ) 2 + 15900 , corresponding to Option B . To break even, the company must sell 600 helmets .
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