35. PROBLEM SOLVING The height of one story of a building is about 10 feet. The bottom of the ladder on the fire truck must be at least 24 feet away from the building. How many stories can the ladder reach? Justify your answer.

To determine how many stories the ladder can reach, we can use the Pythagorean theorem. The ladder, the height of the building, and the distance from the bottom of the ladder to the building form a right triangle. Let's denote: - The height of one story of the building as \( h = 10 \) feet. - The distance from the bottom of the ladder to the building as \( d = 24 \) feet. - The height the ladder can reach as \( x \) stories. According to the Pythagorean theorem, the sum of the squares of the two shorter sides of a right triangle is equal to the square of the hypotenuse. In this case, the hypotenuse is the ladder, and the two shorter sides are the height of the building and the distance from the bottom of the ladder to the building. So, we have the equation: \[ h^2 + d^2 = x^2 \] Substitute the given values: \[ 10^2 + 24^2 = x^2 \] Now, we can solve for \( x \) to find out how many stories the ladder can reach. Calculate the value by following steps: - step0: Calculate: \(10^{2}+24^{2}\) - step1: Evaluate the power: \(100+24^{2}\) - step2: Evaluate the power: \(100+576\) - step3: Add the numbers: \(676\) The sum of the squares of the two shorter sides is 676. Now, we need to find the square root of 676 to determine the height the ladder can reach in feet. Let's calculate this. Calculate the value by following steps: - step0: Calculate: \(\sqrt{676}\) - step1: Write in exponential form: \(\sqrt{26^{2}}\) - step2: Simplify the root: \(26\) The square root of 676 is 26. Therefore, the ladder can reach a height of 26 feet. Since each story of the building is 10 feet tall, the ladder can reach 2.6 stories. So, the ladder can reach approximately 2 stories.