A10-foot ladder leans against a wall with its foot braced 3 feet from wallís base. how far up the wall does the ladder reach?

The answer is √91 feet (which is the same as 9.54 feet).

Imagine this as a right triangle. A length of the foot is actually a hypotenuse (c). The distance from wall's base to the ladder foot is one of the sides of the triangle (let it be a).
So, using the Pythagorean theorem:
c² = a² + b²
It is given:
c = 10 feet
a = 3 feet
b = ?
     c² = a² + b²
⇒ 10² = 3² + b²
    100 = 9 + b²
    b² = 100 - 9 = 91
    ⇒ b = √b² = √91 ≈ 9.54 feet