The following exercise presents a problem similar to an exercise from the previo

Question

The following exercise presents a problem similar to an exercise from the previous section of this chapter. Use Newton's Method to approximate the solution.

You are in a boat a miles from the nearest point on the coast (see figure). You are to go to a point Q, which is b miles down the coast and 1 mile inland. You can row at 3 miles per hour and walk at 4 miles per hour. If a = 3, and b = 4, toward what point on the coast should you row in order to reach Q in the least time? (Round your answer to three decimal places.)

x ____________

Explanation / Answer

minimum distance will be along red dotted line along hypotenuse ,f(x) =sqrt(a^2+x^2) +sqrt((b-x)^2+1^2) including speed we get f(x) =3sqrt(a^2+x^2) +4sqrt((b-x)^2+1) df(x)/dt =3*(2x/2) 1/sqrt(a^2+x^2) +4*(1/2) *2(b-x)(-1)/sqrt((b-x)^2+1) at minimum differentiation will have zero value 3x/sqrt(a^2+x^2) -4(b-x)/sqrt((b-x)^2+1) =0 3x/sqrt(a^2+x^2) =4(b-x) /sqrt( (b-x)^2+1) sqrt(b^2+x^2-2bx+1)/sqrt(a^2+x^2) =(4b-4x)/3x squaring both sides (b^2+x^2-2bx+1)/(a^2+x^2) =(16b^2+16x^2-32bx)/3x^2 3x^2(b^2+x^2-2bx+1)=(16b^2+16x^2-32bx)(a^2+x^2) here a=3 ,b=4 putting values 3x^2(16+x^2-8x+1) =(256+16x^2-128x)(9+x^2) 48x^2+3x^4-24x^3+3x^2 =2304 +144x^2-1152x+256x^2+16x^4-128x^3 45x^2+3x^4-24x^3 =400x^2+16x^4-128x^3-1152x+2304 13x^4-104x^3+355x^2-1152x+2304=0

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