A man of height 1.5 meters walk away from a 5-meter lamppost at a speed of 1.8 m/s. Find the rate at which his shadow is increasing in length.
iwan stands 4 metres away from his school and measured the angle of elevation of the roof to be 67 degrees if Iwan is 1.7 meters tall calculate the height of the school
To find the rate at which the man's shadow is increasing in length, we can use similar triangles and the concept of related rates.
Let's denote the height of the man as h (1.5 meters), the distance between the man and the lamppost as x, and the length of the man's shadow as y.
From the similar triangles formed by the man, his shadow, and the lamppost, we can write the following proportion:
h / y = (h + x) / x
To solve for y in terms of x, we can cross-multiply the proportion:
x * h = y * (h + x)
Expanding this equation:
xh = hy + xy
Rearranging the terms to isolate y:
xy - hy = xh
Factoring out y:
y(x - h) = xh
Dividing both sides by (x - h):
y = xh / (x - h)
Now, we can differentiate both sides of the equation with respect to time (t) to find the rate at which y is changing:
dy/dt = (x * dh/dt + h * dx/dt) / (x - h)
Given that the man's height is constant (since his height is not changing over time), dh/dt = 0. Therefore, the equation simplifies to:
dy/dt = h * dx/dt / (x - h)
Substituting the given values:
dy/dt = 1.5 * 1.8 / (5 - 1.5)
Evaluating the expression:
dy/dt = 2.7 / 3.5
Simplifying:
dy/dt ≈ 0.771 meters per second
Therefore, the rate at which the man's shadow is increasing in length is approximately 0.771 meters per second.
To find the rate at which the man's shadow is increasing in length, we can use similar triangles.
Let's define some variables:
- Let h be the height of the man
- Let x be the distance between the man and his shadow
- Let y be the length of the man's shadow
Initially, when the man is standing right next to the lamppost, the length of his shadow (y) is equal to the height of the lamppost (5 meters), since the sun is directly above the lamppost.
As the man walks away from the lamppost, a similar triangle is formed between the man, his shadow, and the lamppost. The ratio of the corresponding sides of the two similar triangles is equal.
Therefore, we have the following equation:
y / h = (x + y) / h
Simplifying the equation, we get:
y = (x + y) * h / h
y = x + y
Now, we can solve this equation to find the value of x.
x = y - y
x = 0
This means that when the man is right next to the lamppost, his shadow has no length (x = 0).
Now, differentiation both sides of the equation y = x + y with respect to time:
d(y) / dt = d(x) / dt + d(y) / dt
Since the length of the man's shadow (y) is increasing, we can assume that d(y) / dt is the rate at which his shadow is increasing, which is what we need to find.
The value of d(x) / dt is the rate at which the man is moving away from the lamppost, which is given as 1.8 m/s.
d(y) / dt = 1.8 m/s + 0 m/s (since d(x) / dt is 0 as x = 0)
Therefore, the rate at which the man's shadow is increasing in length is 1.8 m/s.
How far away is he?
Otherwise I can only give you the function.
man is p from pole (we need this)
shadow tip is s from pole
we want ds/dt
s/5 = (s-p)/1.5
1.5 s = 5 s - 5 p
3.5 s = 5 p
s = 1.43 p
ds/dt = 1.43 dp/dt
but dp/dt = 1.8
so
ds/dt = 2.57 m/s