a wall 10 feet high is 8 feet from a house. find the length of the shortest ladder that will reach the house when one end rest on the ground outside the wall.

Answer 1

Draw a diagram. Let x be the distance from the wall to the ladder's foot.
Let y be the length of the ladder.
Let θ be the angle the ladder makes with the ground.

tanθ = 10/x
secθ = y/(x+8)

tan^2θ + 1 = sec^2θ

100/x^2 + 1 = y^2/(x+8)^2
or,
y^2 = (x^2+100)(x+8)^2/x^2

y' = 2(x+8)(x^3-100)/x^3
y'=0 when x=-8 or 2∛100 = 9.28

y = 17.28/9.28 * sqrt(9.28^2 + 10^2) = 25.4 feet