An 8 ft wall stands 27 ft from a building. Find the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall.
To solve this problem, we can create a right triangle using the wall, the ground, and the beam.
Let's assume the length of the beam is x feet.
Using the Pythagorean theorem, we can establish the following equation:
x^2 = 8^2 + 27^2
Simplifying the equation, we get:
x^2 = 64 + 729
x^2 = 793
Taking the square root on both sides, we find:
x = √793
Calculating the value, we find that x is approximately 28.14 feet.
Therefore, the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall is approximately 28.14 feet.
To find the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall, we can use the concept of similar triangles.
First, let's draw a diagram to visualize the situation. We have a building, a wall, and a beam that we need to find the length of. We can label the height of the wall as "8 ft" and the distance of the wall from the building as "27 ft".
___________ <- building
| |
| | <- beam
| |
| |
| |
|___________| <- wall
27 ft
Now, let's consider two similar triangles: one formed by the distance between the building and the wall, the height of the wall, and the length of the beam, and another formed by the height of the wall, the length of the beam, and the distance of the beam from the wall.
By setting up a proportion between the two triangles, we can solve for the length of the beam.
The proportion can be set up as follows:
(height of the wall) / (distance of the building from the wall) = (length of the beam) / (distance of the beam from the wall)
Plugging in the given values:
8 ft / 27 ft = (length of the beam) / (distance of the beam from the wall)
To solve for the length of the beam, we can cross-multiply:
(8 ft)(distance of the beam from the wall) = (27 ft)(length of the beam)
Now, we can solve for the length of the beam:
(8 ft)(distance of the beam from the wall) = (27 ft)(length of the beam)
8(distance of the beam from the wall) = 27(length of the beam)
Divide both sides by 8:
distance of the beam from the wall = (27/8)(length of the beam)
Now, we have an equation that relates the distance of the beam from the wall to the length of the beam.
To find the length of the shortest straight beam, we need to minimize the distance of the beam from the wall. In other words, we want to find the smallest possible value for the distance of the beam from the wall.
Since the length of the beam is not given, we cannot find the exact value for the distance of the beam from the wall. However, we know that if the distance of the beam from the wall is 0, the length of the beam will also be 0. Therefore, the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall is 0 ft.
In summary, the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall is 0 ft.
make a sketch.
let x be the distance for the wall to where the beam touches the ground
let the height of the beam along the building be y+8 ft
by similar triangles
y/27 = 8/x
xy = 216 -----> y = 216/x
let L be the length of the beam
L^2 = (8+y)^2 + (27+x)^2
= (8 + 216/x)^2 + (27+x)^2
2L dL/dx = 2(8+216/x)(-216/x^2) + 2(27+x)
= 0 for a min of L
2(8+216/x)(-216/x^2) + 2(27+x) = 0
(8+216/x)(-216/x^2) + (27+x) = 0
-1728/x^2 - 46656/x^3 + 27+x = 0
times x^3
-1728x - 46656 + 27x^3 + x^4 = 0
x^3(x+27) - 1728(x+27) = 0
(x^3 - 1728)(x+27) = 0
x = 12 or x = -27, the last one is a "silly" answer
if x=12
L^2 = (8+18)^2 + (27+12)^2
= 2197
L = √2197 = 46.87