A fence 8 ft. tall runs parallel to a tall building at a distance 4 ft. from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?
Thank you very much! I just have one more question for you. How did you get L = 4/cosA + 8/sinA?
Let L be the ladder length and A be the angle that the ladder makes with the ground. Convince yourself that
L = 4/cosA + 8/sinA
dL/dA = (4/cos^2A)*sinA -(8/sin^2)*cosA
= 0 for minimum ladder length
sin/cos^2A = 2 cosA/sin^2A
sin^3A = 2 cos^3A
tanA = 2^1/3 = 1.260
A = 51.6 deg
Solve for L using the first formula above.
a wall 10 feet high is 8ft from a house. what is the length of the shortest ladder that will reach the house when one end rests on the ground outside the wall?
To find the length of the shortest ladder that will reach from the ground over the fence to the wall of the building, we can use the Pythagorean theorem.
Let's denote the length of the ladder as 'L'. According to the problem, the height of the fence is 8 ft. and the distance between the fence and the building is 4 ft.
We can consider the ladder, the fence, and the distance from the fence to the building as the three sides of a right triangle. The ladder represents the hypotenuse, and the height of the fence and the distance from the fence to the building represent the other two sides.
Applying the Pythagorean theorem, we have:
L^2 = (height of the fence)^2 + (distance from the fence to the building)^2
L^2 = 8^2 + 4^2
L^2 = 64 + 16
L^2 = 80
Taking the square root of both sides, we find:
L = √80
Simplifying the square root, we get:
L ≈ 8.944
Therefore, the length of the shortest ladder that will reach from the ground over the fence to the wall of the building is approximately 8.944 ft.