A certain electronics manufacturer is making GPUs for gaming laptops (GPU stands for graphics processing
unit). The manufacture found that the marginal cost to produce GPUs can be found using the equation
. If the marginal cost were , how many GPUs were produced?
Select an answer

Ask by Vaughn Dunn. in the United States

Mar 25,2025

Question

A certain electronics manufacturer is making GPUs for gaming laptops (GPU stands for graphics processing
unit). The manufacture found that the marginal cost to produce GPUs can be found using the equation
. If the marginal cost were , how many GPUs were produced?
Select an answer

Ask by Vaughn Dunn. in the United States

Mar 25,2025

UpStudy AI · Accepted Answer

We start with the marginal cost function

$$
C = 0.04x^2 - 8x + 800
$$

and we are told that $$ C = 672 $$. Thus, we set up the equation

$$
0.04x^2 - 8x + 800 = 672.
$$

### Step 1. Set up the equation

Subtract 672 from both sides:

$$
0.04x^2 - 8x + 800 - 672 = 0,
$$

which simplifies to

$$
0.04x^2 - 8x + 128 = 0.
$$

### Step 2. Clear the Decimal

Multiply the entire equation by 25 to eliminate the decimal:

$$
25 \cdot (0.04x^2) - 25 \cdot (8x) + 25 \cdot 128 = 0.
$$

This gives:

$$
x^2 - 200x + 3200 = 0.
$$

### Step 3. Solve the Quadratic Equation

Use the quadratic formula

$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
$$

with $$ a = 1 $$, $$ b = -200 $$, and $$ c = 3200 $$. Substitute these into the formula:

$$
x = \frac{-(-200) \pm \sqrt{(-200)^2 - 4(1)(3200)}}{2(1)} = \frac{200 \pm \sqrt{40000 - 12800}}{2}.
$$

Simplify inside the square root:

$$
\sqrt{40000 - 12800} = \sqrt{27200}.
$$

Notice that

$$
27200 = 16 \times 1700 = 16 \times (100 \times 17) = 1600 \times 17,
$$

so

$$
\sqrt{27200} = \sqrt{1600 \times 17} = 40\sqrt{17}.
$$

Thus, the formula becomes

$$
x = \frac{200 \pm 40\sqrt{17}}{2} = 100 \pm 20\sqrt{17}.
$$

### Step 4. Interpret the Solutions

The two solutions are

$$
x = 100 - 20\sqrt{17} \quad \text{or} \quad x = 100 + 20\sqrt{17}.
$$

Approximating these:

- Since $$\sqrt{17} \approx 4.123$$, then
  $$
  100 - 20\sqrt{17} \approx 100 - 82.46 \approx 17.54,
  $$
- and
  $$
  100 + 20\sqrt{17} \approx 100 + 82.46 \approx 182.46.
  $$

Thus, if the marginal cost were $$\$672$$, the function gives two possible production quantities: approximately $$\mathbf{18}$$ GPUs or $$\mathbf{182}$$ GPUs produced.
If the marginal cost is \$672, approximately 18 or 182 GPUs were produced.

A certain electronics manufacturer is making GPUs for gaming laptops (GPU stands for graphics processing
unit). The manufacture found that the marginal cost to produce GPUs can be found using the equation
. If the marginal cost were , how many GPUs were produced?
Select an answer

Solución de inteligencia artificial de Upstudy

Responder

Solución

Step 1. Set up the equation

Step 2. Clear the Decimal

Step 3. Solve the Quadratic Equation

Step 4. Interpret the Solutions

Bonus Knowledge

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Latest Calculus Questions

A certain electronics manufacturer is making GPUs for gaming laptops (GPU stands for graphics processing unit). The manufacture found that the marginal cost to produce GPUs can be found using the equation . If the marginal cost were , how many GPUs were produced? Select an answer

Solución de inteligencia artificial de Upstudy

Responder

Solución

Step 1. Set up the equation

Step 2. Clear the Decimal

Step 3. Solve the Quadratic Equation

Step 4. Interpret the Solutions

Bonus Knowledge

preguntas relacionadas

Latest Calculus Questions

A certain electronics manufacturer is making GPUs for gaming laptops (GPU stands for graphics processing
unit). The manufacture found that the marginal cost to produce GPUs can be found using the equation
. If the marginal cost were , how many GPUs were produced?
Select an answer