Question

a ladder is leaning against a vertical wall. the distance from the top of the ladder to the base of the wall is 21 feet. the distance from the base of the wall to the bottom of the ladder is 6 feet. what is the length of the ladder? provide an answer accurate to the nearest tenth.

Explanation:

Step1: Identify the triangle type

This is a right - triangle problem, so we can use the Pythagorean theorem. The Pythagorean theorem states that for a right - triangle with legs of length \(a\) and \(b\) and hypotenuse of length \(c\), \(c^{2}=a^{2}+b^{2}\). Here, \(a = 21\) (height on the wall) and \(b = 6\) (distance from wall to ladder base), and \(c\) is the length of the ladder.

Step2: Apply the Pythagorean theorem

Substitute \(a = 21\) and \(b = 6\) into the formula \(c=\sqrt{a^{2}+b^{2}}\). First, calculate \(a^{2}\) and \(b^{2}\):
\(a^{2}=21^{2}=441\)
\(b^{2}=6^{2}=36\)
Then, \(a^{2}+b^{2}=441 + 36=477\)

Step3: Calculate the square root

Now, find \(c=\sqrt{477}\approx21.8\) (rounded to the nearest tenth)

Answer:

The length of the ladder is approximately \(21.8\) feet.