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
How many helmets must the company make and sell to break even?
How much will it cost the company to make 600 helmets?
Which system models this situation?
$\begin{array}{l}
y=26 x \text { and } y=8,400(x-500)^2+15,900 \\
y=26 x \text { and } y=-0.030(x-500)^2+15,900 \\
y=x / 26 \text { and } y=-0.030(x-500)^2+15,900 \\
y=x / 26 \text { and } y=8,400(x-500)^2+15,900\end{array}$
Answer (2)
To solve these problems, we need to analyze the information given and determine the appropriate mathematical model to use.
Break-even Point :
The company breaks even when the revenue from selling helmets equals the cost of making them.
Given that each helmet sells for $26, the revenue function is:
y = 26 x
The cost function is a quadratic function with a vertex form:
y = − 0.030 ( x − 500 ) 2 + 15 , 900
To find the break-even point, set these equations equal:
26 x = − 0.030 ( x − 500 ) 2 + 15 , 900
Solving this equation will give the number of helmets where revenue equals cost.
Cost of Making 600 Helmets :
Substitute x = 600 into the cost function:
y = − 0.030 ( 600 − 500 ) 2 + 15 , 900
y = − 0.030 ( 100 ) 2 + 15 , 900
y = − 0.030 ( 10 , 000 ) + 15 , 900
y = − 300 + 15 , 900
y = 15 , 600
So, it will cost the company $15,600 to make 600 helmets.
Choosing the Correct System :
Out of the options given, the correct system that models this situation is:
y = 26 x and y = − 0.030 ( x − 500 ) 2 + 15 , 900
In conclusion, the company must solve the revenue and cost equation to find the break-even point. It costs $15,600 to make 600 helmets, and the correct system of equations to represent this business scenario has been chosen as y = 26 x and y = − 0.030 ( x − 500 ) 2 + 15 , 900 .
The company must find the point at which revenue equals costs to break even. The cost to produce 600 helmets is $15,600, and the correct system of equations modeling this situation is y = 26 x and y = 8 , 400 ( x − 500 ) 2 + 15 , 900 .
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