First, we will start with an understanding of the problem. We have an 8-foot ladder leaning against a wall, such that the top of the ladder is 5.5 feet above the ground. We are asked to find the distance from the bottom of the ladder to the wall.
To solve this, we can use the Pythagorean theorem, which is applicable to right triangles. Recall the Pythagorean theorem:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
Here, [tex]\( c \)[/tex] is the hypotenuse (the length of the ladder), and [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the other two sides of the right triangle. In this scenario:
- [tex]\( c = 8 \)[/tex] feet (length of the ladder)
- [tex]\( a = 5.5 \)[/tex] feet (height from the ground to the point where the ladder touches the wall)
- [tex]\( b \)[/tex] is the distance from the bottom of the ladder to the wall, which we need to find.
Substituting the known values into the Pythagorean theorem:
[tex]\[ (5.5)^2 + b^2 = (8)^2 \][/tex]
First, compute [tex]\( (5.5)^2 \)[/tex] and [tex]\( (8)^2 \)[/tex]:
[tex]\[ (5.5)^2 = 30.25 \][/tex]
[tex]\[ (8)^2 = 64 \][/tex]
Now, substitute these values back into our equation:
[tex]\[ 30.25 + b^2 = 64 \][/tex]
To find [tex]\( b^2 \)[/tex], isolate [tex]\( b^2 \)[/tex] by subtracting 30.25 from both sides:
[tex]\[ b^2 = 64 - 30.25 \][/tex]
[tex]\[ b^2 = 33.75 \][/tex]
To find [tex]\( b \)[/tex], we take the square root of both sides:
[tex]\[ b = \sqrt{33.75} \][/tex]
Therefore, the distance from the bottom of the ladder to the wall is:
[tex]\[ b = \sqrt{33.75} \][/tex]
Hence, the correct answer is:
C [tex]\( \sqrt{33.75} \)[/tex] feet
