Another optimization question

Question

A man is in a boat 2 miles from the nearest point on the coast. He is to go to point Q, located 3 miles down the coast and 1 mile inland (see figure). He can row at a rate of 1 mile per hour and walk at 2 miles per hour. Toward what point on the coast should he row in order to reach point Q in the least time? (Round your answer to two decimal places.) Asks for miles down the coast

Answer 1

Time t =(4 + x²)^1/2/1 + ((3 - x)² + 1)^1/2/2; dt/dx = x/(x² + 4)^1/2 + (x - 3)/(x² - 6x + 10)^1/2 = 0;

x(x² - 6x + 10)^1/2 = (3 - x)(x² + 4)^1/2; x²(x² - 6x + 10) = (3 - x)²(x² + 4); x⁴ - 6x³ + 10x² = x⁴ + 4x² - 6x³ - 24x +

9x² + 36; 6x² + 24x - 36 = 0; x² + 4x - 6 = 0; x = - 2 + √10 = √10 - 2 = 1.162 mi