A ladder is leaning against the wall of a house.The base of the ladder is pulled away from the wall at a rate of 2ft/sec (b) Consider the triangle formed by the side of the house ,theladder, and the ground. Find the rate at which the triangle is changing when the base of the ladder is 7 ft from the wall. (c) Find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is 7 ft from the wall
not enough information, we only have 1 side of the triangle.
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To find the rate at which the triangle is changing, we need to determine the rate at which its height (along the wall) and its base (along the ground) are changing. Let's break down the problem into two parts: (b) finding the rate of change of height and (c) finding the rate of change of the angle.
(b) Finding the rate of change of height:
We can use the Pythagorean theorem to relate the height, base, and ladder length:
height² + base² = ladder²
Differentiating both sides with respect to time (t), we get:
2(height)(d(height)/dt) + 2(base)(d(base)/dt) = 2(ladder)(d(ladder)/dt)
Since we want to find d(height)/dt when the base of the ladder is 7 ft from the wall, we substitute the given value:
2(height)(d(height)/dt) + 2(7)(2) = 2(ladder)(d(ladder)/dt)
Simplifying, we have:
2(height)(d(height)/dt) = 2(ladder)(d(ladder)/dt) - 28
Now, we need to relate the height and the ladder length. Since the ladder is leaning against the wall, we can use similar triangles to determine this relationship:
height/ladder = constant
Let's denote this constant as k:
height = k * ladder
Differentiating both sides with respect to time, we get:
d(height)/dt = k(d(ladder)/dt)
Substituting this into our equation:
2(kladder)(d(ladder)/dt) = 2(ladder)(d(ladder)/dt) - 28
Canceling out the common terms:
2k(d(ladder)/dt) = (d(ladder)/dt) - 14
Now, we can solve for d(ladder)/dt:
2k(d(ladder)/dt) - (d(ladder)/dt) = -14
(2k - 1)(d(ladder)/dt) = -14
d(ladder)/dt = -14 / (2k - 1)
To find the rate at which the triangle is changing when the base of the ladder is 7 ft from the wall, we need to know the value of k.
(c) Finding the rate of change of the angle:
To find the rate at which the angle between the ladder and the wall is changing, we can use trigonometry. Let's denote the angle as θ.
cos(θ) = base/ladder
Differentiating both sides with respect to time, we get:
-sin(θ)(dθ/dt) = (d(base)/dt) / ladder
Substituting the given values (base = 7 ft and ladder = 7 ft from the wall):
-sin(θ)(dθ/dt) = (2 ft/sec) / 7 ft
Now, we can solve for dθ/dt:
dθ/dt = -2 ft/sec / (7 ft sin(θ))