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A ladder leans against a vertical wall. The top of the ladder is 8 meter above the ground. When the bottom of the ladder is moved 2 meter farther away from the wall, the top of the ladder rests against the foot of the wall. What is the length of the ladder?
- a)20 meter
- b)16 meter
- c)18 meter
- d)17 meter
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
A ladder leans against a vertical wall. The top of the ladder is 8 me...
Let the length of the ladder be x meter. We have
82 + y2 = x2 and (y + 2) = x
Hence, 64 + (x - 2)2 = x2
⇒ 64 + x2 — 4x + 4 = x2
⇒ 68 = 4x ⇒ x = 17 meter
Most Upvoted Answer
A ladder leans against a vertical wall. The top of the ladder is 8 me...
Understanding the Problem
To find the length of the ladder, we can use the Pythagorean theorem. Let's denote:
- **L** = length of the ladder
- **h** = height of the ladder above the ground (8 meters)
- **d** = distance from the wall to the base of the ladder
Initially, when the ladder is leaning against the wall:
- **h = 8 m**
- **d = x** (unknown distance from the wall)
According to the Pythagorean theorem:
Equation 1: Initial Position
\[ L^2 = h^2 + d^2 \]
\[
L^2 = 8^2 + x^2 \quad \text{(1)}
\]
When the bottom of the ladder is moved 2 meters away from the wall, the new distance from the wall becomes **(x + 2)**. Now the ladder rests at the foot of the wall, meaning that the height becomes 0.
Equation 2: Final Position
\[ L^2 = h^2 + (x + 2)^2 \]
Since the top of the ladder now touches the foot of the wall, we have:
\[
L^2 = 0^2 + (x + 2)^2 \quad \text{(2)}
\]
Setting the Equations
From Equations (1) and (2):
\[
8^2 + x^2 = (x + 2)^2
\]
Expanding the right side:
\[
64 + x^2 = x^2 + 4x + 4
\]
Now, simplifying:
\[
64 = 4x + 4
\]
\[
60 = 4x
\]
\[
x = 15
\]
Finding the Length of the Ladder
Substituting back to find L:
\[
L^2 = 8^2 + 15^2
\]
\[
L^2 = 64 + 225 = 289
\]
\[
L = \sqrt{289} = 17 \text{ meters}
\]
Thus, the length of the ladder is **17 meters**, which corresponds to option 'D'.
To find the length of the ladder, we can use the Pythagorean theorem. Let's denote:
- **L** = length of the ladder
- **h** = height of the ladder above the ground (8 meters)
- **d** = distance from the wall to the base of the ladder
Initially, when the ladder is leaning against the wall:
- **h = 8 m**
- **d = x** (unknown distance from the wall)
According to the Pythagorean theorem:
Equation 1: Initial Position
\[ L^2 = h^2 + d^2 \]
\[
L^2 = 8^2 + x^2 \quad \text{(1)}
\]
When the bottom of the ladder is moved 2 meters away from the wall, the new distance from the wall becomes **(x + 2)**. Now the ladder rests at the foot of the wall, meaning that the height becomes 0.
Equation 2: Final Position
\[ L^2 = h^2 + (x + 2)^2 \]
Since the top of the ladder now touches the foot of the wall, we have:
\[
L^2 = 0^2 + (x + 2)^2 \quad \text{(2)}
\]
Setting the Equations
From Equations (1) and (2):
\[
8^2 + x^2 = (x + 2)^2
\]
Expanding the right side:
\[
64 + x^2 = x^2 + 4x + 4
\]
Now, simplifying:
\[
64 = 4x + 4
\]
\[
60 = 4x
\]
\[
x = 15
\]
Finding the Length of the Ladder
Substituting back to find L:
\[
L^2 = 8^2 + 15^2
\]
\[
L^2 = 64 + 225 = 289
\]
\[
L = \sqrt{289} = 17 \text{ meters}
\]
Thus, the length of the ladder is **17 meters**, which corresponds to option 'D'.
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A ladder leans against a vertical wall. The top of the ladder is 8 meter above the ground. When the bottom of the ladder is moved 2 meter farther away from the wall, the top of the ladder rests against the foot of the wall. What is the length of the ladder?a)20 meterb)16 meterc)18 meterd)17 meterCorrect answer is option 'D'. Can you explain this answer?
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