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?

• 11.5 feet
• 13.1 feet
• 14.6 feet
• 15.4 feet

Explanation:

Step1: Identify known values

We know the length of the ladder \( L = 15 \) feet and the distance from the base of the wall \( d = 3.5 \) feet. The formula is \( L=\sqrt{d^{2}+h^{2}} \), and we need to solve for \( h \).

Step2: Rearrange the formula for \( h \)

Start with \( L=\sqrt{d^{2}+h^{2}} \). Square both sides to get \( L^{2}=d^{2}+h^{2} \). Then, subtract \( d^{2} \) from both sides: \( h^{2}=L^{2}-d^{2} \). Take the square root of both sides: \( h = \sqrt{L^{2}-d^{2}} \).

Step3: Substitute the known values

Substitute \( L = 15 \) and \( d = 3.5 \) into the formula: \( h=\sqrt{15^{2}-3.5^{2}} \). Calculate \( 15^{2}=225 \) and \( 3.5^{2}=12.25 \). Then, \( 225 - 12.25 = 212.75 \). So, \( h=\sqrt{212.75} \approx 14.6 \) feet.

Answer:

14.6 feet (corresponding to the option "14.6 feet")