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We are given the marginal cost equation for producing $ x $ GPUs as follows:

\[
C'(x) = 0.04x^2 - 7x + 900
\]

The question asks to find how many GPUs were produced when the marginal cost was $865.

To solve this, we set the marginal cost equation equal to 865 and solve for $ x $:

\[
0.04x^2 - 7x + 900 = 865
\]

Let me solve this quadratic equation.The solutions to the equation are approximately $ x = 5.15 $ and $ x = 169.85 $.

Since the number of GPUs produced must be a whole number, and typically you'd round to the nearest integer, the manufacturer produced approximately **5 GPUs** or **170 GPUs**.

Would you like a detailed explanation of how I solved the quadratic equation or have any other questions?

Here are 5 related questions for further practice:
1. How would you calculate the total cost function $ C(x) $ if the marginal cost is known?
2. What is the significance of the marginal cost in production economics?
3. How can we interpret the two possible solutions for $ x $ in this context?
4. How do changes in production scale affect marginal cost?
5. Can the marginal cost ever be negative? Why or why not?

**Tip:** When solving quadratic equations, ensure to check the context of the problem to decide which solution is more appropriate.

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'(x) = 0.04x^2 - 7x + 900. If the marginal cost were $865, how many GPUs were produced?

Learn how to solve a quadratic equation representing the marginal cost for GPU production. This problem involves key algebraic concepts and applies the quadratic formula to determine the number of GPUs produced when the marginal cost is $865. Suitable for Grades 9-12 students.

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