A man 2 meters tall walks at the rate of 2 meters per second toward a streetlight that's 5 meters above the ground. At what rate is the tip of his shadow moving?We've already set this up part of the way. We know that dx/dt = - 2 meters per second, and we're looking for dy/dt.
-10/3
Well, isn't this a shady situation? Let's shed some light on this problem, shall we?
We have a 2-meter tall man walking at a speed of 2 meters per second towards a streetlight that's 5 meters above the ground. We want to find the rate at which the tip of his shadow is moving.
Since we know that the man is moving towards the streetlight, we can assume that his shadow is moving in the opposite direction. So, dx/dt is indeed -2 meters per second.
Now, let's put on our thinking caps and figure out what's happening to the shadow's height, dy/dt.
We can set up a proportion using similar triangles. The height of the man, the height of the streetlight, and the height of the shadow form a right triangle. The ratio of the man's height to his shadow's height is equal to the ratio of the distance between the man and the light source to the height of the light source.
In other words, (2 meters)/(dy/dt) = (2 meters)/(5 meters).
Simplifying this proportion, we find dy/dt = (5 meters)(-2 meters per second)/(2 meters).
After some clownish calculations, we find that the rate at which the tip of his shadow is moving is -5 meters per second.
So, the shadow isn't just creeping along, it's actually moving away pretty quickly from the man. Watch out for disappearing shadows or any other shady business!
To find the rate at which the tip of the man's shadow is moving, we can use similar triangles.
Let's label the height of the man "h" and the length of his shadow "l".
By similar triangles, we know that the ratio of the man's height to his shadow is equal to the ratio of his distance from the streetlight to the length of his shadow:
h/l = (h + 5)/l
Now, let's differentiate both sides of the equation with respect to time (t):
(dh/dt)/l = (h + 5) * (dl/dt)/(l^2)
Given that dx/dt = -2 m/s (the rate at which the man is walking towards the streetlight), we know that dl/dt = dx/dt.
Substituting these values, we have:
(dh/dt)/l = (h + 5) * (-2)/(l^2)
Now, let's solve for (dh/dt):
dh/dt = -2 * (h + 5) * l / (l^2)
Since we know that the man's height is 2 meters, h = 2, and we are given that the length of his shadow is 5 meters, l = 5.
Let's substitute these values into the equation:
dh/dt = -2 * (2 + 5) * 5 / (5^2)
= -2 * 7 * 5 / 25
= -70 / 25
= -2.8 meters per second
Therefore, the rate at which the tip of the man's shadow is moving is -2.8 meters per second.
To find the rate at which the tip of the man's shadow is moving, we need to use similar triangles. Let's set up the problem step by step:
Step 1: Draw a diagram to represent the situation. Label the man's height as "h," the distance between the man and the light source as "x," and the length of the shadow as "y."
Step 2: Express the given information in terms of the variables. We know that the rate at which the man is approaching the light source is given by dx/dt = -2 meters per second (negative because he is moving towards the light source).
Step 3: Write the equation of the similar triangles. The height of the man's shadow is directly proportional to the distance between the man and the light source. Therefore, we have a proportion: (h + y) / x = h / 5.
Step 4: Differentiate the equation with respect to time (t) using the quotient rule. We want to find dy/dt, so we differentiate both sides of the equation with respect to time.
d/dt [(h + y) / x] = d/dt [h / 5]
Step 5: Simplify the differentiation using the chain rule and the product rule.
(dx/dt * (h + y) - (h + y) * dx/dt) / x^2 = 0
Step 6: Substitute the given value of dx/dt and solve for dy/dt.
(-2 * (h + y) - (h + y) * (-2)) / x^2 = 0
Simplifying, we get:
-2h - 2y + 2h + 2y = 0
-2y + 2y = 0
0 = 0
Step 7: Analyze the result. Since we end up with 0 = 0, this means that dy/dt is zero. Therefore, the tip of the man's shadow is not moving.
From what you said, I will assume you defined variables as such....
x is the distance between the man and the streetligh
y is the length of his shadow.
then dx/dt = -2 , find dy/dt
by similar triangles:
2/y = 5/(x+y)
5y = 2x + 2y
3y = 2x
3dy/dt = 2dxdt
dy/dt = 2(-2)/3 = -4/3
so his shadow is decreasing at a rate of 4/3 m/s
(notice , since I said "decreasing", I do not use the negative sign, since the word decreasing implies it)