To find the length of the shortest straight beam that will reach from the ground outside the wall to the side of the building, we need to calculate the hypotenuse of a right triangle, where:

1. The height of the building is the total height we need to reach, i.e., the distance from the ground to the point on the building where the beam will touch.
2. The ground distance (base of the triangle) includes the distance from the wall to the building plus the height of the wall.

Step-by-Step Solution:

1. Define the Variables:

   • Let $h$ represent the height of the wall, which is 6 feet.
   • Let $d$ represent the distance from the wall to the building, which is 28 feet.
   • Let $x$ represent the height on the building where the beam will touch.

   To minimize the length of the beam, we need to find the point on the building that yields the shortest possible beam length.

2. Formulate the Distance Function:

   • The beam forms the hypotenuse of a right triangle, where one leg is $d + h$ (total horizontal distance from the starting point on the ground to the building), and the other leg is $x$ (the height on the building).
   • Using the Pythagorean theorem, the length $L$ of the beam is given by: $L = \sqrt{(d + h)^2 + x^2}$

3. Differentiate and Solve for the Minimum: We set up an equation to find $x$ by differentiating $L$ with respect to $x$ and solving for the critical point.

After following these steps, we find that the shortest length