The marginal cost of a product is modeled by (dC)/(dx) =...

The marginal cost of a product is modeled by $\frac{dC}{dx} = \frac{16}{\sqrt[3]{16x + 7}}$ where $x$ is the number of units. When $x = 17$, $C = 100$. (a) Find the cost function. (Round your constant term to two decimal places.) C = (b) Find the cost of producing 70 units. (Round your answer to two decimal places.) $

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00:01 In this question, we are given the marginal cost of a product, and we are told that the cost of producing 17 products equals to 140.

00:11 And we are asked to find the cost function and the cost of producing 50 units.

00:16 In other words, we are asked to solve the given differential equation.

00:20 And to do that, we are going to multiply both sides of the equation by dx.

00:24 We are going to get d .c equals to 16 over the cube root times dx.

00:34 And then we will differentiate both sides of the equation.

00:40 On the left -hand side, we are going to get c.

00:42 On the right -hand side, we need to calculate the integral.

00:48 And to calculate the integral, we will use u substitution.

00:54 And for u, we are going to choose the expression inside the cube root.

01:00 So, u equals to 16x plus 1.

01:05 From that equation, du equals to 16 times dx, which is very fortunate because 16 dx, x is already inside the integral, so we can replace it by du.

01:22 And then, in terms of u, the integral becomes du, divided by the cube root of u.

01:33 We can rewrite that as the integral of u to the negative 1 3rd power, du, and then use the power rule.

01:43 By the power rule, we need to add 1 to the exponent and divide by the new exponent...

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