A man of height 1.7 meters walks away from a 5-meter lamppost at a speed of
1.6 m/s. Find the rate at which his shadow is increasing in length. (Round your answer to three decimal places.) This problem is absolutely killing me I see a lot of examples like this on here but I can not seem to follow the steps correctly nor understand what is going on.
0.824 m/s
I figured it out hopefully this helps some of you. So draw your picture out and label the distance between the man and the pole as Y and the distance between the man and the shadow tip as X. Now use the properties of similar triangles to set up a proportion and then cross multiply.
5/(Y+X)=1.7/X
5X=1.7Y+1.7X
3.3X=1.7Y
Then use we need to find dx/dt! So start differentiating then isolate dx/dt.
3.3 (dx/dt)=1.7 (dy/dt)
(dx/dt)=(1.7(dy/dt))/3.3
Now just plug in the value they gave for the mans speed, this happens to be dy/dt.
The answer comes out to be 0.824 m/s.
Hope that helps!
Ah, the age-old question of shadows and their ever-increasing lengths. Fear not, my friend, for I am here to shed some light on this matter with my clownish wisdom. Let's sashay through the steps together and tackle this problem once and for all.
First, let's define some variables:
h = height of the man (1.7 meters)
x = distance between the man and the lamppost
y = length of the shadow
We are given that the man is walking away from the lamppost at a speed of 1.6 m/s. This means that the distance x between the man and the lamppost is changing over time. Since we want to find the rate at which the shadow is increasing, we need to find dy/dt (the rate of change of y with respect to time).
To start, let's set up a triangle to represent the situation:
/|
h / |
/ |
/ | y
/ |
/-----|----|
x y
Now, by similar triangles, we can set up a ratio between the lengths of the sides:
h/y = x/(x+y)
Let's differentiate both sides of this equation with respect to time t:
(d/dt)(h/y) = (d/dt)(x/(x+y))
Using the quotient rule, we get:
(h(dy/dt) - y(dh/dt))/y^2 = [(x+y)(dx/dt) - x(dy/dt)]/(x+y)^2
Since dh/dt represents the rate at which the man's height is changing, and the problem doesn't mention anything about that, we can assume it's zero (unless he's growing or shrinking, which would be quite peculiar!).
Plugging in the values we know: h = 1.7m, dx/dt = -1.6 m/s (because he's walking away from the lamppost):
(1.7(dy/dt) - y(0))/y^2 = [(x+y)(-1.6) - x(dy/dt)]/(x+y)^2
Simplifying, we get:
(1.7(dy/dt))/y^2 = (-1.6(x+y))/(x+y)^2
Canceling out the common factors and rearranging, we end up with:
(1.7/y^2) dy/dt = -1.6/(x+y)
Now, let's plug in the remaining known values: y = 5m, x = (x+y)-y = 5m-1.7m = 3.3m.
(1.7/5^2) dy/dt = -1.6/(3.3+5)
(1.7/25) dy/dt = -1.6/8.3
Solving for dy/dt, we find:
dy/dt = (1.7/25) * (-1.6/8.3)
Now, grab a calculator and perform this simple calculation to get the rate at which the man's shadow is increasing. Remember to round it to three decimal places for that extra pizzazz!
I hope this helps brighten your understanding of the problem, and remember, when shadows get longer, it's never a bad time for a proper clown show to brighten the mood!
To solve this problem, we can use similar triangles. Let's denote the height of the man as h, the distance between the man and the lamppost as d, and the length of his shadow as s.
We are given that the height of the man is 1.7 meters and the rate at which he is walking away from the lamppost is 1.6 m/s. We need to find the rate at which his shadow is increasing, which is ds/dt.
Using similar triangles, we can set up the following relationship:
h / s = d / (d + s)
We take the derivative of both sides with respect to time (t) to find the rates of change:
(d/dt)(h/s) = (d/dt)(d / (d + s))
To find dh/dt, we need to know the rate at which the man's height is changing. Since this information is not given, we can assume that the height of the man is constant. Thus, dh/dt is 0.
Now, let's differentiate the right side of the equation using the quotient rule:
(d/dt)(d / (d + s)) = [(d + s)(ddt) - d(d/dt)(d + s)] / (d + s)^2
Since the man is walking away from the lamppost, ds/dt is positive. The derivative of d with respect to time, ddt, is the rate at which the man is moving away from the lamppost, which is 1.6 m/s.
Plugging in the known values into the equation, we have:
0 = [(5 + s)(1.6) - 5(ds/dt)] / (5 + s)^2
Solving for ds/dt, we can rewrite the equation:
[(5 + s)(1.6)] = 5(ds/dt)
8 + 1.6s = 5(ds/dt)
ds/dt = (8 + 1.6s) / 5
Now, we can substitute the value of s when the man is at a certain distance from the lamppost to calculate the rate at which his shadow is increasing. For example, if we want to find the rate at which his shadow is increasing when he is 3 meters away from the lamppost, we substitute s = 3 into the equation:
ds/dt = (8 + 1.6(3)) / 5
ds/dt = 14.8 / 5
ds/dt = 2.96 m/s
Therefore, the rate at which his shadow is increasing in length when he is 3 meters away from the lamppost is 2.96 m/s.
To solve this problem, we can use similar triangles. Let's start by identifying the triangles involved:
1. The first triangle is formed by the man's height, the length of his shadow, and the distance between him and the lamppost.
- Let's call the man's height h, the length of his shadow x, and the distance between him and the lamppost d.
2. The second triangle is formed by the man's shadow, the length of the lamppost, and the distance between him and the lamppost.
- Let's call the length of the lamppost L and the length of the man's shadow s.
Now, let's find the relationship between the lengths of the sides of both triangles using similar triangles:
The ratio of the lengths in similar triangles is consistent. So, we have:
h / x = L / (x + s)
Now, let's differentiate this equation with respect to time (t), since we need to find the rate at which the man's shadow is increasing.
Differentiating both sides of the equation, we get:
dh/dt * (x + s) - dx/dt * L = 0
Since we need to find the rate at which the shadow is increasing (ds/dt), we rearrange the equation to solve for ds/dt:
ds/dt = (dh/dt * (x + s)) / L
Now, let's substitute the given values into the equation:
h = 1.7 m (height of the man)
d = 5 m (distance between the man and the lamppost)
dx/dt = 1.6 m/s (speed at which the man is walking)
L = 5 m (length of the lamppost)
x = ?
s = ?
To find x and s, we can use the Pythagorean theorem:
x^2 + h^2 = d^2
s^2 + L^2 = (d + x)^2
Solving these equations simultaneously will give us the values of x and s.
Once we have the values of x and s, we can plug them into ds/dt = (dh/dt * (x + s)) / L to calculate the rate at which the man's shadow is increasing (ds/dt). Round the answer to three decimal places.
Okay. If the man is x meters from the pole, and his shadow's length is s meters, then using similar triangles, we have
(x+s)/5 = s/1.7
So, that means that
x = 5s/1.7 - s
or, s = 0.51x
So, ds/dt = 0.51 dx/dt = 0.51*1.6 = 0.82 m/s