A man of height 1.8m walks away from a lamp at a height of 6m. If the man's speed is 7m/s, find the speed in m/s at which the tip of the shadow moves.
h=1.8 m, H=6 m, v=7 m/s u=?
x is the distance of man to lamp post,
y is the distance from the tip of the shadow to the lamp post.
From the similar triangles
H/h = y/(y-x),
6/1.8 = y/(y-x),
6y -6x=1.8y,
4.2 y=6 x,
0.7y=x,
d{0.7y}/dt=dx/dt,
0.7(dy/dt)=dx/dt,
0.7u =v,
u=v/0.7=7/0.7=10 m/s.
draw a right triangle: vertical 6m, horizontal leg x
Now within that triangle, another vertical 1.8 m, and the horizontal distance to the lamppost is L, so the length to the tip of shadow is x-L
similar triangles:
x/6=(x-L)/1.8
or 1.8x-6x=-6L
4.2 dx/dt=6dL/dt
given: dL/dt=7, solve for dx/dt
To find the speed at which the tip of the shadow moves, we can use the concept of similar triangles. Let's denote the speed at which the tip of the shadow moves as Vs.
From the given information, we have the following set-up:
Height of the man: h1 = 1.8m
Height of the lamp: h2 = 6m
Speed of the man: Vm = 7m/s
We can set up a proportion using the similar triangles formed by the man, the shadow, and the lamp. The proportion is as follows:
(h1 + h2) / h2 = (Vm + Vs) / Vs
Substituting the given values, we have:
(1.8 + 6) / 6 = (7 + Vs) / Vs
Simplifying the left side of the equation, we get:
7.8 / 6 = (7 + Vs) / Vs
Cross-multiplying, we have:
7.8 * Vs = 6 * (7 + Vs)
Simplifying further:
7.8 * Vs = 42 + 6Vs
Rearranging the terms:
7.8 * Vs - 6Vs = 42
Combining like terms:
1.8 * Vs = 42
Dividing both sides by 1.8:
Vs = 42 / 1.8
Calculating the value:
Vs ≈ 23.33 m/s
Therefore, the speed at which the tip of the shadow moves is approximately 23.33 m/s.
To find the speed at which the tip of the shadow moves, we need to consider the similar triangles formed by the man, the lamp, and the shadow.
Let's denote the distance of the man from the lamp as "x" (measured horizontally from the lamp), and the length of the shadow as "s".
Since the triangles formed by the man, the lamp, and the shadow are similar, we can set up the following proportion:
(man's height) / (lamp's height) = (shadow's length) / (distance from the lamp to the man)
Using the given values, we can write the equation as:
1.8 / 6 = s / x
To find the speed at which the tip of the shadow moves, we need to differentiate this equation with respect to time.
Differentiating both sides with respect to time (t), we get:
(1.8/6) * dx/dt = ds/dt
dx/dt represents the rate at which the man is moving away from the lamp, which is given as 7 m/s.
Substituting the values, we have:
(1.8/6) * 7 = ds/dt
Simplifying the equation, we get:
ds/dt = (1.8/6) * 7 = 2.1 m/s
Therefore, the speed at which the tip of the shadow moves is 2.1 m/s.