Shortest ladder—more realistic An 8-ft-tall fence runs parallel to the wall of a house at a distance of 5 ft. Find the length of the shortest ladder that extends from the ground, over the fence, to the house. Assume that the vertical wall of the house is 20 ft high and the horizontal ground extends 20 ft from the fence. Further Explorations and Applications

Solution Step 1: The ladder position minimizes the ladder length.the objective function in this problem is the ladder length L.The position of the ladder is specified by x the distance between the foot of the ladder and fence.The goal is to express the function of x where x>0. Step 2: We can draw the graph from the given data Step 3: The pythagorean theorem gives the relationship L = (5 + x) + b 2 Here b is the height of the top of the ladder above the ground ABE and DCE are similar triangles Similar triangles gives the constraint x= x+5 b = 8(x+5) x Step 4: Solve the constraint equation for b and express L in terms of x 2 L = (5 + x) + ( 8(x+5) x L = (x + 5) (1 + 64) x2