A company plans to sell bicycle helmets for $26 each. The company’s business manager estimates that the cost, , of making helmets is a quadratic function with a -intercept of 8,400 and a vertex of .
number of helmets
amount in dollars

How many helmets must the company make and sell to break even?
helmets
COMPLETE
How much will it cost the company to make 600 helmets?
$
DONE

Which system models this situation?

Ask by Thornton Schneider. in the United States

Mar 22,2025

Question

A company plans to sell bicycle helmets for $26 each. The company’s business manager estimates that the cost, , of making helmets is a quadratic function with a -intercept of 8,400 and a vertex of .
number of helmets
amount in dollars

How many helmets must the company make and sell to break even?
helmets
COMPLETE
How much will it cost the company to make 600 helmets?
$
DONE

Which system models this situation?

Ask by Thornton Schneider. in the United States

Mar 22,2025

UpStudy AI · Accepted Answer

1. Write the vertex form of the cost function. Since the vertex is $$(500,15900)$$, the cost function can be written as  
$$
y=-0.03(x-500)^2+15900
$$  
where the value of $$a$$ is found as follows:
   - The vertex form is $$\displaystyle y=a(x-500)^2+15900$$.
   - Use the $$y$$-intercept $$(0,8400)$$:
     $$
     8400=a(0-500)^2+15900 \quad\Rightarrow\quad 8400=a\cdot250000+15900.
     $$
   - Solve for $$a$$:
     $$
     a\cdot250000=8400-15900=-7500 \quad\Rightarrow\quad a=-\frac{7500}{250000}=-0.03.
     $$

2. Determine the break-even point. The company earns revenue by selling helmets at \$26 each. The revenue function is  
$$
y=26x.
$$
Break-even occurs when revenue equals cost:
$$
26x=-0.03(x-500)^2+15900.
$$
It is given that break-even happens at $$x=600$$ helmets. We verify:
   - Revenue at $$x=600$$:
     $$
     26(600)=15600.
     $$
   - Cost at $$x=600$$:
     $$
     -0.03(600-500)^2+15900=-0.03(100^2)+15900=-0.03(10000)+15900= -300+15900=15600.
     $$
Since both equal \$15,600, break-even occurs when $$600$$ helmets are made and sold.

3. Find the cost to make 600 helmets. From the cost function:
$$
y=-0.03(600-500)^2+15900=-0.03(100^2)+15900=-300+15900=15600.
$$
Thus, the cost is $$\$15600$$.

4. The system that models this situation is:
$$
\begin{cases}
y=26x\[1mm]
y=-0.03(x-500)^2+15900
\end{cases}
$$
To break even, the company must make and sell 600 helmets. The cost to make 600 helmets is \$15,600. The system that models this situation is: $$ \begin{cases} y=26x\ y=-0.03(x-500)^2+15900 \end{cases} $$

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