To determine the height [tex]\( h \)[/tex] that a 15-foot ladder reaches on a wall when it is placed 3.5 feet from the base of the wall, we can use the Pythagorean theorem. The Pythagorean theorem states that for a right triangle with a hypotenuse [tex]\( L \)[/tex], one leg [tex]\( d \)[/tex], and the other leg [tex]\( h \)[/tex], the following relationship holds:
[tex]\[ L^2 = d^2 + h^2 \][/tex]
We are given:
- [tex]\( L = 15 \)[/tex] feet (length of the ladder)
- [tex]\( d = 3.5 \)[/tex] feet (distance from the base of the wall)
We need to solve for [tex]\( h \)[/tex]. Rearranging the Pythagorean theorem to solve for [tex]\( h \)[/tex]:
[tex]\[ h^2 = L^2 - d^2 \][/tex]
[tex]\[ h = \sqrt{L^2 - d^2} \][/tex]
First, we calculate [tex]\( L^2 \)[/tex] and [tex]\( d^2 \)[/tex]:
[tex]\[ L^2 = 15^2 = 225 \][/tex]
[tex]\[ d^2 = 3.5^2 = 12.25 \][/tex]
Next, we substitute these values back into the equation for [tex]\( h \)[/tex]:
[tex]\[ h = \sqrt{225 - 12.25} \][/tex]
[tex]\[ h = \sqrt{212.75} \][/tex]
Upon calculating the square root of 212.75, we find:
[tex]\[ h \approx 14.586 \][/tex]
Therefore, the ladder reaches a height of approximately 14.6 feet on the wall. From the given options, the closest correct height is:
- 11.5 feet
- 13.1 feet
- 14.6 feet (Correct Answer)
- 15.4 feet
So, the height that the 15-foot ladder reaches on the wall when placed 3.5 feet from the base is approximately 14.6 feet.
