A man 1.8m tall walks away from a lamp post 4m high at a speed of 1.5 m/s. How fast is his shadow lengthen ?
When he is x m from the pole, and his shadow is s m long, we have
s/1.8 = (x+s)/4
ds/dt = 1/4 (dx/dt + ds/dt)
3 ds/dt = 1.5
ds/dt = 0.5 m/s
To find how fast the man's shadow lengthens, we can use similar triangles.
Let's assume that the length of the man's shadow is x meters, and the distance between the man and the lamp post is y meters. We can set up the following proportion:
Height of the man / Height of the lamp post = Length of the shadow / Distance between the man and the lamp post
1.8m / 4m = x / y
Cross-multiplying the proportion:
1.8m * y = 4m * x
y = (4m * x) / 1.8m
Simplifying:
y = (4/1.8) * x
y = 2.222 * x
Now, let's differentiate this equation with respect to time (t) since we want to find how fast the shadow lengthens. We'll use the chain rule of differentiation:
d/dt (y) = d/dt (2.222 * x)
The derivative of y with respect to t (dy/dt) will give us the rate of change of the shadow length:
dy/dt = 2.222 * dx/dt
We know that the man is walking away from the lamp post at a speed of 1.5 m/s, so dx/dt = 1.5 m/s.
Substituting this value:
dy/dt = 2.222 * 1.5 m/s
Calculating the result:
dy/dt = 3.333 m/s
Therefore, the man's shadow lengthens at a rate of 3.333 meters per second.
To find out how fast the man's shadow is lengthening, we need to use similar triangles and the concept of proportional relationships.
Let's denote the height of the man as Hm (1.8m) and the height of the lamp post as Hp (4m). The length of the man's shadow can be represented as Xm, and we want to find how fast Xm is changing with respect to time.
Since the man is walking away from the lamp post, we can assume that the distance between the man and the lamp post is changing at a constant rate. Let's call this distance D.
Now, let's examine the similar triangles formed by the man, his shadow, and the lamp post.
In the first triangle, we have:
Height of the man / Height of the lamp post = Length of the man's shadow / Distance between the man and the lamp post
Hm / Hp = Xm / D
Next, let's take the derivative of this equation with respect to time (t) to find the rate of change of Xm:
d/dt (Hm / Hp) = d/dt (Xm / D)
Since the height of the man does not change over time, we can simplify the equation to:
0 = d/dt (Xm / D)
Now, let's solve for dXm/dt, which represents the rate of change of Xm (the length of the man's shadow) over time. We can rearrange the equation as follows:
dXm/dt = 0 * D
Since the right side of the equation is zero and D is a constant, we conclude that the rate at which the man's shadow lengthens is zero. This means that the length of the shadow remains constant as the man walks away from the lamp post at a constant speed.
Therefore, the answer to the question is that the man's shadow length does not change or lengthen at all.