# Solve using Newton's Method

#### beastcoast515

Marcus Tool and Die Company produces a specialized milling tool designed specifically for machining ceramic components. Each milling tool sells for \$4, so the company's revenue in dollars for x units sold is R(x) = 4x. The company's cost in dollars to produce x tools can be modeled as C(x) = 304+30x^(5/8). Use Newton's method to find the break-even point for the company (that is, find x such that C(x) = R(x)). Use x = 370 as your initial guess and show all your work. Thank you!

If this turns out to be an elaborate multi-step problem, please let me know so that I can compensate you for your time via PayPal. Again, I appreciate the help.

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#### Shakarri

Set up the equation describing the problem
$$\displaystyle C(x)=R(x)$$

$$\displaystyle 304+30x^{5/8}=4x$$

$$\displaystyle 304+30x^{5/8}-4x=0$$

let f(x) be
$$\displaystyle f(x)=304+30x^{5/8}-4x$$

the derivative of this is
$$\displaystyle f'(x)=\frac{150}{8}x^{-3/8}-4$$

When your first guess (x1) is 370 the second estimate (x2) will be

$$\displaystyle x_2=370-\frac{f(370)}{f'(370)}$$

$$\displaystyle 390-\frac{304+30(370)^{5/8}-4(370)}{\frac{150}{8}(370)^{-3/8}-4}$$

When you have a second estimate repeat the procedure but replace 390 with your second estimate.