A building is to be braced by means of a beam which must pass over a wall. If the wall is 3 3/8 feet high and stands 8 feet from the building, find the shortest beam that can be used.

if the beam touches the ground x feet from the wall, and touches the building at a height of y feet, then

x/3.375 = (x+8)/y
y = (27/8)(x+8)/x

The length z of the beam is

z^2 = x^2+y^2 = x^2 + ((27/8)(x+8)/x)^2

Crank through the algebra, and you find that dz/dx=0 at x≈5.9736

So,
y = 7.8949
z = √(13.9736^2 + 7.8949^2) ≈16 feet

Well, the shortest beam that can be used is definitely not the one that passes under the wall. That would be a pretty "low" point of view, you know? So, let's wrap our minds around a different option.

Since we want the shortest beam, let's think outside the box. How about we try to brace the building by using a beam that goes through the wall? I mean, who needs walls to be intact when you can have a beam sticking right through it, am I right?

So, let's calculate the length of the beam that goes through the wall. We have a wall that is 3 3/8 feet high and stands 8 feet from the building. This creates a right triangle, with the beam acting as the hypotenuse. We can use the Pythagorean theorem to find the length of the beam.

Using a^2 + b^2 = c^2, where a and b are the legs of the triangle and c is the hypotenuse, we can plug in the numbers:

a^2 + b^2 = c^2
(3 3/8)^2 + 8^2 = c^2

Calculating the squaring and adding, we get:

(27/8)^2 + 64 = c^2
729/64 + 64 = c^2
729/64 + 4096/64 = c^2
(729 + 4096)/64 = c^2
4825/64 = c^2

Taking the square root of both sides, we find:

sqrt(4825/64) = c
69.5281/8 = c
8.691 d'oh! I mean, c = 8.691!

So, the shortest beam that can be used to brace the building is approximately 8.691 feet. Voilà! We have our answer, and a slightly unconventional but humorous approach to it!

To find the shortest beam that can be used, we can use the Pythagorean Theorem. Let's follow these steps:

Step 1: Draw a diagram to visualize the problem. Create a right triangle with the wall as the vertical leg, the distance between the wall and the building as the horizontal leg, and the shortest beam as the hypotenuse.

Step 2: Identify the given values:
- Height of the wall = 3 3/8 feet
- Distance between wall and building = 8 feet

Step 3: Calculate the length of the hypotenuse (shortest beam) using the Pythagorean Theorem:
c² = a² + b²

Let's assume the height of the wall (a) is the vertical leg, and the distance between the wall and the building (b) is the horizontal leg.
a = 3 3/8 feet = 3.375 feet
b = 8 feet

Substitute the values into the Pythagorean Theorem equation:
c² = (3.375)² + 8²
c² = 11.390625 + 64
c² = 75.390625

Take the square root of both sides to find c:
c = √(75.390625)
c ≈ 8.68 feet

Step 4: Round the answer to the nearest hundredth:
The shortest beam length that can be used is approximately 8.68 feet.

To find the shortest beam that can be used, we need to consider the height of the wall and the distance from the building.

Let's break down the problem step by step:

1. Draw a diagram: Draw a diagram to visualize the situation. Label the building, the wall, and the distance between them.

2. Determine the total height: To find the total height that the beam needs to cover, we need to consider the height of the wall and the height of the building. Let's add them up:
Total Height = Height of Wall + Height of Building

Given that the height of the wall is 3 3/8 feet, we can convert it to a mixed fraction:
Height of Wall = 3 + 3/8 = 24/8 + 3/8 = 27/8 feet

Since we don't have information about the height of the building, let's assume it to be h feet.

Total Height = 27/8 + h feet

3. Determine the length of the beam: To find the length of the beam, we need to calculate the distance between the building and the wall. Given that the distance 8 feet.

4. Use the Pythagorean Theorem: To find the shortest beam length, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

In our case, the hypotenuse is the length of the beam, and the other two sides are the total height and the distance between the building and the wall.

Length of Beam^2 = Total Height^2 + Distance^2

Substitute the values we found:
Length of Beam^2 = (27/8 + h)^2 + 8^2

We can simplify the equation further, but we won't need to find the exact value since we only want the shortest beam length. Thus, we can stop here with the equation.

5. Find the shortest beam length: Since we do not have the value of h, we cannot find the exact length of the beam without additional information or assumptions. However, we can assume different values for h and calculate the corresponding beam lengths.

By varying the value of h, we can calculate multiple beam lengths and determine the shortest one.

Therefore, to find the shortest beam, you need to use the Pythagorean theorem and make assumptions or obtain additional information about the height of the building.