Question
A company plans to sell bicycle helmets for
each. The company’s business manager estimates
that the cost, y, of making
function with a
of (
each. The company’s business manager estimates
that the cost, y, of making
function with a
of (
Ask by Barrett Logan. in the United States
Mar 22,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The cost function is
Solution
To find the quadratic function representing the cost
where
-
Identify the vertex:
-
-
Substitute the vertex into the vertex form:
-
Use the y-intercept to find
:
The y-intercept is given as. This occurs when . We can substitute and into the equation to find : Simplifying this gives:Rearranging to solve for: -
Substituting
back into the equation:
Now we can write the complete quadratic function:
This function represents the cost
Next, we can expand this equation to standard form if needed. Let’s do that:
Now, let’s simplify this expression.
Simplify the expression by following steps:
Simplify the expression by following steps:
- step0: Solution:
- step1: Expand the expression:
- step2: Add the numbers:
The expanded form of the quadratic function representing the costof making helmets is:
This function can be used to analyze the cost of producing different quantities of helmets.
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution Bonus Knowledge
To break even, the total revenue from selling the helmets must equal the total cost of producing them. Since each helmet sells for
Looking at this situation, one common mistake is forgetting to consider both the fixed costs and variable costs when formulating your break-even point. Be mindful that if your revenue doesn’t cover fixed costs upon increasing production, you might still be in the red, even with a high number of units sold. Always calculate and graph both functions to better visualize where they intersect!
