A company plans to sell bicycle helmets for
each. The company’s business manager estimates
that the cost, y, of making helmets is a quadratic to break even?
function with a -intercept of 8,400 and a vertex the company make and
of ( ).

Ask by Barrett Logan. in the United States

Mar 22,2025

Question

A company plans to sell bicycle helmets for
each. The company’s business manager estimates
that the cost, y, of making helmets is a quadratic to break even?
function with a -intercept of 8,400 and a vertex the company make and
of ( ).

Ask by Barrett Logan. in the United States

Mar 22,2025

UpStudy AI · Accepted Answer

To find the quadratic function representing the cost $$ y $$ of making $$ x $$ helmets, we can use the vertex form of a quadratic equation, which is given by:

$$
y = a(x - h)^2 + k
$$

where $$ (h, k) $$ is the vertex of the parabola. In this case, the vertex is $$ (500, 15900) $$.

1. **Identify the vertex**: 
   - $$ h = 500 $$
   - $$ k = 15900 $$

2. **Substitute the vertex into the vertex form**:
   $$
   y = a(x - 500)^2 + 15900
   $$

3. **Use the y-intercept to find $$ a $$**:
   The y-intercept is given as $$ 8400 $$. This occurs when $$ x = 0 $$. We can substitute $$ x = 0 $$ and $$ y = 8400 $$ into the equation to find $$ a $$:

$$
   8400 = a(0 - 500)^2 + 15900
   $$

Simplifying this gives:
   $$
   8400 = a(250000) + 15900
   $$

Rearranging to solve for $$ a $$:
   $$
   8400 - 15900 = 250000a
   $$
   $$
   -7500 = 250000a
   $$
   $$
   a = \frac{-7500}{250000}
   $$
   $$
   a = -0.03
   $$

4. **Substituting $$ a $$ back into the equation**:
   Now we can write the complete quadratic function:
   $$
   y = -0.03(x - 500)^2 + 15900
   $$

This function represents the cost $$ y $$ of making $$ x $$ helmets.

Next, we can expand this equation to standard form if needed. Let's do that:

$$
y = -0.03(x^2 - 1000x + 250000) + 15900
$$

Now, let's simplify this expression.
Simplify the expression by following steps:
- step0: Solution:
  $$-0.03\left(x^{2}-1000x+250000\right)+15900$$
- step1: Expand the expression:
  $$-0.03x^{2}+30x-7500+15900$$
- step2: Add the numbers:
  $$-0.03x^{2}+30x+8400$$
The expanded form of the quadratic function representing the cost $$ y $$ of making $$ x $$ helmets is:

$$
y = -0.03x^{2} + 30x + 8400
$$

This function can be used to analyze the cost of producing different quantities of helmets.
The cost function is $$ y = -0.03x^{2} + 30x + 8400 $$.

A company plans to sell bicycle helmets for
each. The company’s business manager estimates
that the cost, y, of making helmets is a quadratic to break even?
function with a -intercept of 8,400 and a vertex the company make and
of ( ).

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A company plans to sell bicycle helmets for each. The company’s business manager estimates that the cost, y, of making helmets is a quadratic to break even? function with a -intercept of 8,400 and a vertex the company make and of ( ).

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Latest Algebra Questions

A company plans to sell bicycle helmets for
each. The company’s business manager estimates
that the cost, y, of making helmets is a quadratic to break even?
function with a -intercept of 8,400 and a vertex the company make and
of ( ).