a man climbing a ladder which is inclined to the wall at an angle of 30 degree. if he ascend at the rate of 2m/s then he approach the wall at the rate of
To find the rate at which the man approaches the wall, we can use trigonometry.
Given:
Angle of inclination (θ) = 30 degrees
Rate of ascent (v) = 2 m/s
We need to find the rate at which the man approaches the wall. Let's call it "r".
Using trigonometry, we can relate the rate of ascent, rate of approach, and the angle of inclination:
Sin(θ) = r / v
Substituting the values:
Sin(30) = r / 2
Sin(30) is equal to 0.5, so we have:
0.5 = r / 2
To isolate "r", we can multiply both sides of the equation by 2:
2 * 0.5 = r
1 = r
Therefore, the man approaches the wall at a rate of 1 meter per second.
To find the rate at which the man approaches the wall, we need to determine the horizontal component of his climbing speed. Since the ladder is inclined at an angle of 30 degrees, we can use trigonometry to find this horizontal component.
The horizontal component of the climbing speed is given by the formula: horizontal speed = climbing speed * cos(angle of inclination)
Given that the climbing speed is 2m/s and the angle of inclination is 30 degrees, we can substitute these values into the formula:
horizontal speed = 2m/s * cos(30 degrees)
To find the value of cos(30 degrees), you can either use a calculator or consult a trigonometric table. The cosine of 30 degrees is √3/2 or approximately 0.866.
horizontal speed = 2m/s * 0.866
Now we can calculate the horizontal speed:
horizontal speed = 1.732 m/s
Therefore, the man approaches the wall at a rate of 1.732 meters per second.
since cot 30 = √3, the ratio of the horizontal distance x to the height h is
x/h = √3
so, if the height is increasing at 2 m/s, the man approaches the wall at 2√3 m/s