To determine how high up the wall the ladder reaches, we can use the Pythagorean theorem. The Pythagorean theorem relates the lengths of the sides of a right triangle and is stated as:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
In this scenario:
- [tex]\( c \)[/tex] is the length of the ladder (the hypotenuse),
- [tex]\( a \)[/tex] is the distance from the base of the ladder to the wall (one leg of the triangle),
- [tex]\( b \)[/tex] is the height up the wall the ladder reaches (the other leg of the triangle).
Given:
- The length of the ladder ([tex]\( c \)[/tex]) is 15 feet,
- The distance from the base of the ladder to the wall ([tex]\( a \)[/tex]) is 9 feet.
We need to find the height ([tex]\( b \)[/tex]) the ladder reaches up the wall.
1. Substitute the known values into the Pythagorean theorem:
[tex]\[ 9^2 + b^2 = 15^2 \][/tex]
2. Calculate the squares of the known values:
[tex]\[ 81 + b^2 = 225 \][/tex]
3. Solve for [tex]\( b^2 \)[/tex] by subtracting 81 from both sides:
[tex]\[ b^2 = 225 - 81 \][/tex]
[tex]\[ b^2 = 144 \][/tex]
4. Find [tex]\( b \)[/tex] by taking the square root of both sides:
[tex]\[ b = \sqrt{144} \][/tex]
[tex]\[ b = 12 \][/tex]
Thus, the height up the wall the ladder reaches is:
B. 12 ft.
