To determine the number of GPUs produced when the marginal cost is [tex]$748$[/tex], we need to solve the given equation for [tex]\( x \)[/tex] (number of GPUs produced). The given equation is:
[tex]\[ C = 0.04x^2 - 5x + 900 \][/tex]
We are told that the marginal cost [tex]\( C \)[/tex] is [tex]$748. Thus, we substitute \( 748 \) for \( C \):
\[ 748 = 0.04x^2 - 5x + 900 \]
Next, we move all terms to one side of the equation to set it to zero:
\[ 0.04x^2 - 5x + 900 - 748 = 0 \]
Simplify the equation:
\[ 0.04x^2 - 5x + 152 = 0 \]
This is a quadratic equation of the form \( ax^2 + bx + c = 0 \), where \( a = 0.04 \), \( b = -5 \), and \( c = 152 \).
To solve this quadratic equation, we use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Plugging in the values of \( a \), \( b \), and \( c \):
\[ x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(0.04)(152)}}{2(0.04)} \]
\[ x = \frac{5 \pm \sqrt{25 - 24.32}}{0.08} \]
\[ x = \frac{5 \pm \sqrt{0.68}}{0.08} \]
Calculate the square root of \( 0.68 \):
\[ \sqrt{0.68} \approx 0.8246 \]
Now substitute this value back into the equation:
\[ x = \frac{5 \pm 0.8246}{0.08} \]
This gives us two solutions:
\[ x = \frac{5 + 0.8246}{0.08} \approx \frac{5.8246}{0.08} \approx 72.807 \]
\[ x = \frac{5 - 0.8246}{0.08} \approx \frac{4.1754}{0.08} \approx 52.192 \]
Therefore, the manufacturer produced approximately 52.192 or 72.807 GPUs when the marginal cost was $[/tex]748. The exact solutions to the number of GPUs produced are:
[tex]\[ x \approx 52.192 \quad \text{or} \quad x \approx 72.807 \][/tex]
Thus, the number of GPUs produced when the marginal cost is $748 is:
[tex]\[ x = 52.192 \quad \text{or} \quad x = 72.807 \][/tex]
