Question

a 14 foot ladder is placed 5 feet from the wall. how high up does the ladder reach? round to the nearest hundredth if necessary.

Explanation:

Step1: Identify the problem type

This is a right triangle problem where the ladder is the hypotenuse, the distance from the wall is one leg, and the height on the wall is the other leg. We use the Pythagorean theorem: \(a^2 + b^2 = c^2\), where \(c = 14\) (ladder length), \(b = 5\) (distance from wall), and we solve for \(a\) (height).

Step2: Rearrange the formula

We need to find \(a\), so rearrange the Pythagorean theorem: \(a = \sqrt{c^2 - b^2}\)

Step3: Substitute the values

Substitute \(c = 14\) and \(b = 5\) into the formula: \(a = \sqrt{14^2 - 5^2}\)

Step4: Calculate the squares

Calculate \(14^2 = 196\) and \(5^2 = 25\), so \(a = \sqrt{196 - 25}\)

Step5: Subtract inside the square root

\(196 - 25 = 171\), so \(a = \sqrt{171}\)

Step6: Calculate the square root

\(\sqrt{171} \approx 13.08\) (rounded to the nearest hundredth)

Answer:

13.08