A fence 5 feet tall runs parallel to a tall building at a distance of 7 feet 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? [Hint: Determine the length of a ladder that touches the building, fence, and ground as a function of the acute angle the ladder makes with the ground.]
make a sketch showing the ladder touching the fence and making contact with the wall.
let the angle at the base of the ladder be ß
draw a horizontal from the top of the fence to the wall, then the angle the ladder makes with that line is ß, giving up 2 similar right-angled triangles
the ladder has length L1 above the fence and L2 below the fence
cosß = 7/L1 ---> L1 = 7secß
sinß = 5/L2 ---> L2 = 5cscß
so L = 7secß + 5sinß = 7(cosß)^-1 + 5(sinß)^-1
dL/dß = -7(cosß)^-2(-sinß) - 5(sinß)^-2(cosß)
simplifying and setting equal to zero gives me
7sinß/(cosß)^2 = 5cosß/(sinß)^2
cross-multiplying:
7(sinß)^3 = 5(cosß)^3 or
(tanß)^3 = 5/7
tanß = (5/7)^(1/3) = .8939..
ß=41.7936 degrees
sub that back into L1 and L2, add them up , ....
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 trigonometry.
Let's call the height of the building "h" and the distance from the base of the building to the fence "d".
We have a right triangle formed by the ladder, the ground, and the building. The ladder is the hypotenuse, and the height of the building is one of the legs. The other leg is the distance from the base of the building to the fence. Let's call the length of the ladder "L".
Using the Pythagorean theorem, we have:
L^2 = h^2 + d^2
Substituting the given values:
L^2 = 5^2 + 7^2
= 25 + 49
= 74
Taking the square root of both sides:
L = √74
Therefore, the length of the shortest ladder that will reach from the ground over the fence to the wall of the building is √74 feet.
To determine the length of the shortest ladder that will reach from the ground over the fence to the wall of the building, we need to use trigonometry. Let's denote the height of the building as 'h', the distance of the fence from the building as 'd', and the length of the ladder as 'L'.
We can create a right triangle with the ladder as the hypotenuse, the vertical side as the height of the building, and the horizontal side as the distance of the fence from the building. Since the ladder touches the ground, it forms another right triangle with the horizontal side as the distance of the fence from the building and the vertical side as the height of the fence.
Using trigonometric functions, we can find the relationship between the sides of the triangle. Let's use the sine function:
sin(angle) = opposite/hypotenuse
In the first right triangle, the opposite side is the height of the building 'h' and the hypotenuse is the length of the ladder 'L'. So, we have:
sin(angle) = h/L
In the second right triangle, the opposite side is the height of the fence, which is 5 feet, and the hypotenuse is the length of the ladder 'L'. So, we have:
sin(angle) = 5/L
Since the angle is the same in both triangles, we can equate the two expressions:
h/L = 5/L
Now, we can solve this equation for the length of the ladder 'L':
h = 5
L = (5/h) * L
L = (5/7) * L
Therefore, the length of the shortest ladder that will reach from the ground over the fence to the wall of the building is (5/7) times the height of the building.