Solve using Newton's Method

Dec 2012
New Hampshire
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.
Oct 2012
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.