dtan5457
  • dtan5457
pre calc question
Mathematics
jamiebookeater
  • jamiebookeater
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dtan5457
  • dtan5457
anonymous
  • anonymous
You want to minimize the TOTAL TIME. Split that into two parts: Tw = walking time. Tr = rowing time. Now make a diagram. He is at M, and wants to reach Q. So he aims the boat at P: | + Q(3,1) | | | P(x,0) -+-------+--*----+-------+----- Coast | 1 2 3 | | | M(0,-2) MP = sqrt(x^2 + 4) PQ = sqrt((x-3)^2 + 1) Row MP in Tr = MP/2 = sqrt(x^2 + 4)/2 Walk PQ in Tw = PQ/4 = sqrt((x-3)^2 + 1)/4 Total time is the sum of those; T = sqrt(x^2 + 4)/2 + sqrt((x-3)^2 + 1)/4 Now do the usual stuff: 2x 2(x - 3) dT/dx = ---------------- + -------------------- 4 sqrt(x^2 + 4) 8 sqrt((x-3)^2 + 1) x (x - 3) dT/dx = ---------------- + -------------------- 2 sqrt(x^2 + 4) 4 sqrt((x-3)^2 + 1) Set that equal to zero: x (x - 3) 0 = ---------------- + -------------------- 2 sqrt(x^2 + 4) 4 sqrt((x-3)^2 + 1) 2x sqrt((x-3)^2 + 1) + (x - 3)sqrt(x^2 + 4) 0 = ----------------------------------------... 4 sqrt(x^2 + 4) sqrt((x-3)^2 + 1) Set the top to zero: 2x sqrt((x-3)^2 + 1) + (x - 3)sqrt(x^2 + 4) = 0 2x sqrt((x-3)^2 + 1) = (3 - x)sqrt(x^2 + 4) Square both sides: 4x^2((x - 3)^2 + 1) = (9 - 6x + x^2)(x^2 + 4) 4x^2(x^2 - 6x + 9 + 1) = (9 - 6x + x^2)(x^2 + 4) 4x^4 - 24x^3 + 40x^2 = 9x^2 - 6x^3 + x^4 + 36 - 24x + 4x^2 3x^4 - 18x^3 + 27x^2 = + 36 - 24x 3x^4 - 18x^3 + 27x^2 + 24x - 36 = 0 x^4 - 6x^3 + 9x^2 + 8x - 12 = 0 x = 1 is an obvious solution: 1 - 6 9 8 - 12 (1) 1 -5 4 12 ------------------------ 1 - 5 4 12 zero x^3 - 5x^2 + 4x + 12 = 0 is the reduced equation. A quick-and-dirty graph of x^3 - 5x^2 + 4x + 12 shows only a root for a negative number. So x = 1 is the solution
Here_to_Help15
  • Here_to_Help15
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