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)
Substitute the given values L = 15 and d = 3.5 into the formula L = d 2 + h 2 .
Square both sides to get 1 5 2 = ( 3.5 ) 2 + h 2 , which simplifies to 225 = 12.25 + h 2 .
Solve for h 2 : h 2 = 225 − 12.25 = 212.75 .
Take the square root to find h : h = 212.75 ≈ 14.6 feet. The final answer is 14.6 .
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 .
Substituting Values Substitute the given values into the formula: 15 = ( 3.5 ) 2 + h 2 .
Squaring Both Sides Square both sides of the equation to get rid of the square root: 1 5 2 = ( 3.5 ) 2 + h 2 .
Calculating Squares Calculate the squares: 225 = 12.25 + h 2 .
Isolating h squared Solve for h 2 : h 2 = 225 − 12.25 = 212.75 .
Finding h Solve for h by taking the square root of both sides: h = 212.75 .
Calculating the Height Calculate the square root: h ≈ 14.58595 . 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 isn't just a math problem; it's crucial for safety in construction and everyday tasks. For example, firefighters use similar calculations to determine the safe distance to place a ladder from a building, ensuring they can reach the necessary height while maintaining stability. Similarly, when setting up a ladder to clean gutters or paint a house, knowing the correct distance from the wall ensures the ladder won't slip or fall. This simple formula helps prevent accidents and ensures tasks are completed safely and efficiently.
To find the height a 15-foot ladder reaches when placed 3.5 feet from a wall, we use the formula L = d 2 + h 2 . After calculations, we find that the height h is approximately 14.6 feet. Therefore, the answer is C. 14.6 feet.
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