The formula that relates the length of a ladder, [tex]$L$[/tex], that leans against a wall with distance [tex]$d$[/tex] from the base of the wall and the height [tex]$h$[/tex] that the ladder reaches up the wall is [tex]$L=\sqrt{d^2+h^2}$[/tex]. What height on the wall will a 15-foot ladder reach if it is placed 3.5 feet from the base of a wall?
A. 11.5 feet
B. 13.1 feet
C. 14.6 feet
D. 15.4 feet
Answer (2)
Using the Pythagorean theorem, a 15-foot ladder placed 3.5 feet from the wall reaches approximately 14.6 feet up the wall. Thus, the answer is C. 14.6 feet.
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Substitute the given values into the formula: 15 = 3. 5 2 + h 2 .
Square both sides: 1 5 2 = 3. 5 2 + h 2 .
Solve for h 2 : h 2 = 1 5 2 − 3. 5 2 = 225 − 12.25 = 212.75 .
Take the square root: h = 212.75 ≈ 14.6 . The height is 14.6 feet.
Explanation
Understanding the Problem We are given the formula L = d 2 + h 2 , where L is the length of the ladder, d is the distance from the base of the wall, and h is the height the ladder reaches on the wall. We are given that L = 15 feet and d = 3.5 feet. We want to find the height h that the ladder reaches on the wall.
Substituting the Values Substitute the given values of L and d into the formula: 15 = 3. 5 2 + h 2 .
Squaring Both Sides Square both sides of the equation to get: 1 5 2 = 3. 5 2 + h 2
Isolating h^2 Solve for h 2 : h 2 = 1 5 2 − 3. 5 2
Calculating h^2 Calculate h 2 : h 2 = 225 − 12.25 = 212.75
Finding h Take the square root of both sides to find h : h = 212.75
Calculating h and Rounding Calculate the value of h : h ≈ 14.585952145814822 Rounding to one decimal place, we get h ≈ 14.6 feet.
Final Answer Therefore, the height on the wall that the 15-foot ladder will reach is approximately 14.6 feet.
Examples
Understanding how ladders lean against walls involves the Pythagorean theorem, a fundamental concept in geometry. This principle is not just for ladders; it's used in construction to ensure buildings are square, in navigation to calculate distances, and even in art to create perspective. For example, architects use this to calculate roof slopes, ensuring structural integrity and proper water runoff. Similarly, carpenters rely on it for precise measurements when building furniture, guaranteeing stability and aesthetic appeal. The Pythagorean theorem's applications are vast, making it a cornerstone of both practical and creative endeavors.
