A man is walking beside a 30-m street lamp. After walking 25 m away from the street lamp, he sunk into a quicksand. At the time when 2 m of him are above the ground, he is sinking at 1 m/s. At what rate is the length of his shadow changing at that moment?
if his shadow has length s, and his height is x meters, then we have
s/x = (s+25)/30
or,
s = 25x/(30-x)
ds/dt = 750/(30-x)^2 dx/dt
so, when x=2,
ds/dt = 750/28^2 (-1) = -0.9566 m/s
Can any one help me to answer?
To find the rate at which the length of his shadow is changing, we can use similar triangles and the concept of related rates. Let's denote the length of the man's shadow as x, and the distance between the man and the street lamp as y.
We are given that the man is walking 25 m away from the street lamp, so y = 25 m. We are also given that the man is sinking into quicksand at a rate of 1 m/s when 2 m of him are above the ground.
Since the man is sinking into quicksand, the distance between his head and the ground is decreasing at a rate of 1 m/s. We can represent this as dy/dt = -1 m/s.
Let's set up a proportion using the similar triangles formed by the man, his shadow, and the street lamp:
x/y = (x + 2)/(y + 30)
Differentiate both sides of the equation with respect to time (t) using the chain rule:
(d/dt)(x/y) = (d/dt)((x + 2)/(y + 30))
To find the rate at which the length of his shadow is changing, we need to solve for dx/dt. Rearranging the equation:
(dx/dt * y - x * dy/dt) / y^2 = (1/(y + 30)) * (dx/dt * (y + 30) - (x + 2) * dy/dt)
Substituting the given values:
( dx/dt * 25 - x * (-1) ) / 25^2 = (1/55) * ( dx/dt * (25 + 30) - (x + 2) * (-1) )
Simplifying the equation:
25dx/dt + x = (55/25) ( 55 * dx/dt - x - 2)
Simplifying further:
25dx/dt + x = 55dx/dt - (55/25) * (x + 2)
Combining like terms:
25dx/dt - 55dx/dt = (55/25) * (x + 2) - x
Multiplying both sides by 25:
25( -30dx/dt ) = (55/25) * (x + 2) - x
Simplifying again:
-30dx/dt = (55/25)x + (110/25) - x
Combining like terms:
-30dx/dt = (30/25)x + (110/25)
Simplifying further:
-30dx/dt = (6/5)x + (22/5)
Dividing both sides by -30:
dx/dt = -((6/5)x + (22/5))/30
Now we substitute the value of x = 25 into the equation to find the rate at which the length of his shadow is changing when 2m of him are above the ground:
dx/dt = -((6/5)(25) + (22/5))/30
Calculating the equation:
dx/dt = -((30 + 22)/5)/30
Simplifying:
dx/dt = -(52/5)/30
dx/dt ≈ -0.867 m/s
Therefore, the rate at which the length of his shadow is changing at that moment is approximately -0.867 m/s. The negative sign indicates that the shadow is getting shorter.