Question

a company produces two types of bicycles, a and b. the cost of producing each type is given by the two functions $c_a(x)=0.05x^{2}+50x + 1000$ and $c_b(x)=0.03x^{2}+70x + 2000$, where $x$ represents the number of hours worked on the assembly line. nicole is asked to determine the number of hours worked on the assembly line where the cost functions are the same. determine if nicoles work is correct or incorrect and justify your decision. for any portion of nicoles response in which there is an error, provide and explain an alternative response. if you wish to include an image in your explanation, you can draw the image on paper, take a picture, and insert it into your response using the image tool.

Explanation:

Step1: Set the cost - functions equal

Set $C_A(x)=C_B(x)$. So, $0.05x^{2}+50x + 1000=0.03x^{2}+70x + 2000$.

Step2: Rearrange the equation

Subtract $0.03x^{2}+70x + 2000$ from both sides:
\[

$$\begin{align*} 0.05x^{2}+50x + 1000-(0.03x^{2}+70x + 2000)&=0\\ 0.05x^{2}+50x + 1000 - 0.03x^{2}-70x - 2000&=0\\ (0.05x^{2}-0.03x^{2})+(50x - 70x)+(1000 - 2000)&=0\\ 0.02x^{2}-20x - 1000&=0 \end{align*}$$

\]
Multiply through by 100 to clear the decimals: $2x^{2}-2000x - 100000 = 0$. Divide by 2: $x^{2}-1000x - 50000=0$.

Step3: Use the quadratic formula

The quadratic formula for $ax^{2}+bx + c = 0$ is $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Here, $a = 1$, $b=-1000$, and $c=-50000$.
\[

$$\begin{align*} x&=\frac{-(-1000)\pm\sqrt{(-1000)^{2}-4\times1\times(-50000)}}{2\times1}\\ &=\frac{1000\pm\sqrt{1000000 + 200000}}{2}\\ &=\frac{1000\pm\sqrt{1200000}}{2}\\ &=\frac{1000\pm1095.45}{2} \end{align*}$$

\]
We get two solutions:
$x_1=\frac{1000 + 1095.45}{2}=\frac{2095.45}{2}=1047.725$ and $x_2=\frac{1000 - 1095.45}{2}=\frac{-95.45}{2}=-47.725$.
Since the number of hours $x$ cannot be negative, the number of hours worked on the assembly - line when the cost functions are the same is approximately $x = 1047.73$ hours.

To determine if Nicole's work is correct or not, we would need to see Nicole's work. But the above is the correct way to solve the problem of finding when the two cost functions are equal.

Answer:

The number of hours $x$ when the cost functions are equal is approximately $1047.73$ hours (assuming Nicole's work is not provided for comparison). If Nicole's work is provided, we would compare her steps and results with the above - shown correct solution process.