To determine how high the ladder reaches on the wall, you can use the Pythagorean theorem. The Pythagorean theorem is applicable here because the ladder, the distance from the wall to the base of the ladder, and the height at which the ladder touches the wall form a right triangle.
The Pythagorean theorem states:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
Here:
- [tex]\( c \)[/tex] is the length of the ladder, which is the hypotenuse of the right triangle.
- [tex]\( a \)[/tex] is the distance from the base of the ladder to the wall.
- [tex]\( b \)[/tex] is the height the ladder reaches on the wall, which is what we need to find.
Given:
- The length of the ladder [tex]\( c \)[/tex] is 4 meters.
- The distance from the base of the ladder to the wall [tex]\( a \)[/tex] is 1.2 meters.
We need to find [tex]\( b \)[/tex], the height the ladder reaches on the wall.
First, rewrite the Pythagorean theorem to solve for [tex]\( b \)[/tex]:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
[tex]\[ b^2 = c^2 - a^2 \][/tex]
[tex]\[ b = \sqrt{c^2 - a^2} \][/tex]
Next, substitute the given values:
[tex]\[ b = \sqrt{4^2 - 1.2^2} \][/tex]
[tex]\[ b = \sqrt{16 - 1.44} \][/tex]
[tex]\[ b = \sqrt{14.56} \][/tex]
[tex]\[ b \approx 3.8 \][/tex]
Thus, the height the ladder reaches on the wall is approximately 3.8 meters, rounded to one decimal place.
