Question

the formula that relates the length of a ladder, l, that leans against a wall with distance d from the base of the wall and the height h that the ladder reaches up the wall is $l = \sqrt{d^2 + h^2}$. what height on the wall will a 15-foot ladder reach if it is placed 3.5 feet from the base of a wall?
\bigcirc\\ 11.5 feet
\bigcirc\\ 13.1 feet
\bigcirc\\ 14.6 feet
\bigcirc\\ 15.4 feet

Explanation:

Step1: Substitute known values

We know \( L = 15 \) (length of the ladder) and \( d = 3.5 \) (distance from the base of the wall). Substitute these into the formula \( L=\sqrt{d^{2}+h^{2}} \), we get \( 15=\sqrt{(3.5)^{2}+h^{2}} \).

Step2: Square both sides

To eliminate the square root, square both sides of the equation: \( 15^{2}=(\sqrt{(3.5)^{2}+h^{2}})^{2} \), which simplifies to \( 225 = 12.25+h^{2} \).

Step3: Solve for \( h^{2} \)

Subtract \( 12.25 \) from both sides: \( h^{2}=225 - 12.25=212.75 \).

Step4: Solve for \( h \)

Take the square root of both sides: \( h=\sqrt{212.75}\approx14.6 \) (rounded to one decimal place).

Answer:

14.6 feet