Let's solve the problem step by step to determine how many GPUs were produced when the marginal cost is \$442.
Given the equation for the marginal cost:
[tex]\[ C = 0.04x^2 - 7x + 700 \][/tex]
We are told that the marginal cost \( C \) is \$442, so we set up the equation:
[tex]\[ 0.04x^2 - 7x + 700 = 442 \][/tex]
Next, we need to solve for \( x \). First, we subtract 442 from both sides of the equation to set it to zero:
[tex]\[ 0.04x^2 - 7x + 700 - 442 = 0 \][/tex]
Simplify the equation:
[tex]\[ 0.04x^2 - 7x + 258 = 0 \][/tex]
This is a quadratic equation in standard form \( ax^2 + bx + c = 0 \), where \( a = 0.04 \), \( b = -7 \), and \( c = 258 \). Now, we solve this quadratic equation.
Using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), we find the roots of the equation.
The values are:
[tex]\[ a = 0.04 \][/tex]
[tex]\[ b = -7 \][/tex]
[tex]\[ c = 258 \][/tex]
Plug these values into the quadratic formula:
[tex]\[ x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(0.04)(258)}}{2(0.04)} \][/tex]
[tex]\[ x = \frac{7 \pm \sqrt{49 - 41.28}}{0.08} \][/tex]
[tex]\[ x = \frac{7 \pm \sqrt{7.72}}{0.08} \][/tex]
Calculate the two potential solutions for \( x \):
[tex]\[ \sqrt{7.72} \approx 2.78 \][/tex]
So,
[tex]\[ x_1 = \frac{7 + 2.78}{0.08} \approx \frac{9.78}{0.08} \approx 122.23 \][/tex]
and
[tex]\[ x_2 = \frac{7 - 2.78}{0.08} \approx \frac{4.22}{0.08} \approx 52.77 \][/tex]
Thus, the two possible values for \( x \) are approximately 122.23 and 52.77.
Therefore, the number of GPUs produced when the marginal cost is \$442 could be approximately 52.77 or 122.23 (since we typically consider whole numbers for actual items produced, it may be useful in practical contexts to round these results). But for the purpose of this problem, both values are considered.
