A note card company has found that the marginal cost per card of producing \( x \) note cards is given by the function below, where c'( \( x \) ) is the marginal cost, in cents, per card. Find the total cost of producing 760 cards, disregarding any fixed costs. \( C^{\prime}(x)=-0.03 x+86 \), for \( x \leq 1000 \) The total cost is \( \square \) cents

To find the total cost of producing 760 cards, we need to integrate the marginal cost function \( C'(x) \) with respect to \( x \) to obtain the total cost function \( C(x) \). Given: \( C'(x) = -0.03x + 86 \) for \( x \leq 1000 \) To find the total cost function \( C(x) \), we integrate \( C'(x) \) with respect to \( x \): \[ C(x) = \int C'(x) \, dx \] Substitute \( C'(x) = -0.03x + 86 \) into the integral: \[ C(x) = \int (-0.03x + 86) \, dx \] Now, we can integrate the function to find the total cost function \( C(x) \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int -0.03x+86 dx\) - step1: Use properties of integrals: \(-\int 0.03x dx+\int 86 dx\) - step2: Evaluate the integral: \(-\frac{3}{200}x^{2}+86x\) - step3: Add the constant of integral C: \(-\frac{3}{200}x^{2}+86x + C, C \in \mathbb{R}\) The total cost function \( C(x) \) is given by: \[ C(x) = -\frac{3}{200}x^{2} + 86x + C \] where \( C \) is a constant of integration. To find the total cost of producing 760 cards, we need to substitute \( x = 760 \) into the total cost function \( C(x) \) and calculate the result. Substitute \( x = 760 \) into the total cost function: \[ C(760) = -\frac{3}{200}(760)^{2} + 86(760) + C \] Now, we can calculate the total cost of producing 760 cards. Simplify the expression by following steps: - step0: Solution: \(\frac{-3}{200}\times 760^{2}+86\times 760+C\) - step1: Rewrite the fraction: \(-\frac{3}{200}\times 760^{2}+86\times 760+C\) - step2: Multiply the numbers: \(-8664+86\times 760+C\) - step3: Multiply the numbers: \(-8664+65360+C\) - step4: Add the numbers: \(56696+C\) The total cost of producing 760 cards is \( 56696 + C \) cents, where \( C \) is a constant of integration. Since we are disregarding any fixed costs, the constant of integration \( C \) is 0. Therefore, the total cost of producing 760 cards is \( 56696 \) cents.