To solve this problem, we need to use the Pythagorean theorem, which relates the three sides of a right-angled triangle. In this case, the ladder forms the hypotenuse of the right triangle, the distance from the wall to the foot of the ladder is one leg, and the height that the ladder reaches on the wall is the other leg.
Given:
- Length of the ladder, [tex]\( L = 15 \)[/tex] feet
- Distance from the base of the wall, [tex]\( d = 3.5 \)[/tex] feet
We need to find the height [tex]\( h \)[/tex] that the ladder reaches on the wall.
1. First, we use the Pythagorean theorem's formula:
[tex]\[ L^2 = d^2 + h^2 \][/tex]
2. Plug in the values we know:
[tex]\[ 15^2 = 3.5^2 + h^2 \][/tex]
3. Calculate [tex]\( 15^2 \)[/tex] and [tex]\( 3.5^2 \)[/tex] :
[tex]\[ 225 = 12.25 + h^2 \][/tex]
4. Subtract [tex]\( 12.25 \)[/tex] from both sides to isolate [tex]\( h^2 \)[/tex]:
[tex]\[ 225 - 12.25 = h^2 \][/tex]
[tex]\[ 212.75 = h^2 \][/tex]
5. Take the square root of both sides to solve for [tex]\( h \)[/tex]:
[tex]\[ h = \sqrt{212.75} \][/tex]
6. When finding the square root of 212.75, we get approximately:
[tex]\[ h \approx 14.585952145814822 \][/tex]
So, the height on the wall that the ladder reaches when placed 3.5 feet from the base of the wall is approximately 14.6 feet.
Thus, the correct answer is:
[tex]\[ \boxed{14.6 \text{ feet}} \][/tex]
