A foot ladder is leaning against a wall so that the bottom o

A foot ladder is leaning against a wall so that the bottom of the ladder is 5 feet from the wall (see diagram). You want to determine how much farther from the wall the bottom of the lad must be moved so that the top of the ladder is exactly 10 feet above the floor. a. The new ladder position forms another right triangle in which x represents the additional distance the foot of the ladder has moved from the wall. Use the Pythagorean theorem to write an equation relating the base, height, and hypotenuse of this new right triangle. b. Solve the equation from part a to determine x.

Solution

The ladder forms a right angled triangle with the wall being the perpendicular to the floor, and the ladder being the hypotenuse. When shifted, the hypotenuse is 16ft., the perpendicular is 10 ft., and base is x+5 ft. (say). As per the Pythagoras theorem, we have 16² = 10²+ (x+5)² , where x ft. is the additional distance by which the ladder is shifted to the right. Thus, (x+5)² = 256-100 = 156. Therefore, (x+5) = 156 = 12.49 so that x = 12.49-5 = 7.49 ft. Thus, the ladder should be shifted to the right by 12.49 ft. so that the top of the ladder is 10 ft. above the floor.

(a). The required equation is 16² = 10²+ (x+5)² , where x ft. is the additional distance by which the ladder is shifted to the right. The equation, on simplification is, (x+5)² = 156.

(b). If (x+5)² = 156, then x+5 = 156 = 12.49 so that x = 12.49-5 = 7.49 ft.( on rounding off to 2 decimal places).