Questions: A certain electronics manufacturer is making GPUs for gaming laptops (GPU stands for graphics processing unit). The manufacturer found that the marginal cost C to produce x GPUs can be found using the equation C=0.03 x^2-8 x+700. If the marginal cost were 190, how many GPUs were produced?

A certain electronics manufacturer is making GPUs for gaming laptops (GPU stands for graphics processing unit). The manufacturer found that the marginal cost C to produce x GPUs can be found using the equation C=0.03 x^2-8 x+700. If the marginal cost were 190, how many GPUs were produced?
Transcript text: EXERCISES 3.4 Derivatives as Rates of Change Progres Score: 4/6 Answered: 4/6 Question 5 A certain electronics manufacturer is making GPUs for gaming laptops (GPU stands for graphics processing unit). The manufacture found that the marginal cost $C$ to produce $\times$ GPUs can be found using the equation $C=0.03 x^{2}-8 x+700$. If the marginal cost were $\$ 190$, how many GPUs were produced? Select an answer $\vee$ Question Help: Video Submit Question
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Solution

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Solution Steps

Step 1: Define the Equation

We start with the marginal cost equation given by the manufacturer: \[ C = 0.03x^{2} - 8x + 700 \] We set \( C = 190 \) to find the number of GPUs produced: \[ 0.03x^{2} - 8x + 700 = 190 \]

Step 2: Rearrange the Equation

Rearranging the equation gives us: \[ 0.03x^{2} - 8x + 510 = 0 \]

Step 3: Identify Coefficients

From the quadratic equation \( ax^{2} + bx + c = 0 \), we identify the coefficients:

  • \( a = 0.03 \)
  • \( b = -8 \)
  • \( c = 510 \)
Step 4: Calculate the Discriminant

We calculate the discriminant \( D \) using the formula: \[ D = b^{2} - 4ac \] Substituting the values, we find: \[ D = (-8)^{2} - 4(0.03)(510) = 64 - 61.2 = 2.8 \]

Step 5: Solve for \( x \)

Since the discriminant is positive, we can find the two solutions for \( x \) using the quadratic formula: \[ x = \frac{-b \pm \sqrt{D}}{2a} \] Substituting the values: \[ x = \frac{8 \pm \sqrt{2.8}}{2(0.03)} \]

Step 6: Simplify the Solutions

Calculating the two possible values for \( x \): \[ x_1 = \frac{8 + \sqrt{2.8}}{0.06} \] \[ x_2 = \frac{8 - \sqrt{2.8}}{0.06} \]

These values represent the number of GPUs produced when the marginal cost is $190.

Final Answer

\(\boxed{x \approx 133.33 \text{ or } 126.67}\)

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