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?
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 $\vee$
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Solution
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:
\[
0.03x^{2} - 8x + 700 = 190
\]
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.