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{a^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 into the formula: 15 = 3. 5 2 + h 2 .
Square both sides: 225 = 12.25 + h 2 .
Solve for h 2 : h 2 = 212.75 .
Take the square root and round to one decimal place: h = 212.75 ≈ 14.6 .
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 the length of the ladder is 15 feet, so L = 15 , and the distance from the base of the wall is 3.5 feet, so d = 3.5 . We need 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 L = d 2 + h 2 to get 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 ) 2 , which simplifies to 225 = 3. 5 2 + h 2 .
Calculating 3. 5 2 Calculate 3. 5 2 = 3.5 × 3.5 = 12.25 . So, the equation becomes 225 = 12.25 + h 2 .
Isolating h 2 Subtract 12.25 from both sides of the equation to isolate 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 the Square Root Calculate the square root of 212.75: h = 212.75 ≈ 14.58595 . Round the value of h to one decimal place: 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 instance, architects use this theorem to design stable structures, and carpenters rely on it for precise measurements when building furniture. The relationship L = d 2 + h 2 helps ensure safety and accuracy in various real-world applications.
The height on the wall that a 15-foot ladder reaches when placed 3.5 feet from its base is approximately 14.6 feet. This is calculated using the Pythagorean theorem. Therefore, the correct answer is C. 14.6 feet.
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