A man 1.69 meters tall is walking towards a building at the rate of 1.5 meters per second. If there is a light on the ground 15 meter from the building, how fast is the man's shadow on the building growing shorter when he is 5 meters from the building.
If the man's shadow has length s, then using similar triangles, when the man is x meters from the building,
s/15 = 1.69/(15-x)
s = 25.35/(15-x)
ds/dt = 25.25/(15-x)^2 dx/dt
Now plug in your numbers
Well, it seems like our good ol' shadow problem! Let me crack a joke before we dive into calculations.
Why don't shadows ever go to scary movies?
Because they're already "shade"ing themselves! 😄
Now, let's tackle this problem. We have a man walking towards a building, and we need to figure out how fast his shadow on the building is getting smaller.
First, let's draw a diagram to visualize the situation. The man is 1.69 meters tall, and the light is 15 meters away from the building. When the man is 5 meters from the building, he will create a right triangle with the shadow and light as the other two sides.
Using the Pythagorean theorem, we can determine the length of the man's shadow when he is 5 meters from the building.
(Shadow length)^2 + (Distance from the light)^2 = (Distance from the building)^2
Let x be the length of the shadow, so we have:
x^2 + 15^2 = 5^2
Simplifying the equation will give us the value of x, which represents the length of the shadow.
Now, to find how quickly the shadow is changing, we need to take the derivative of the equation with respect to time and solve for dx/dt.
But don't worry, I'll spare you all those technical details, because I have a better idea. Let's take a break from the math and enjoy another joke!
Why did the shadow bring a ladder to college?
Because it wanted to get a higher education! 😂
All right, back to business. After you've found the length of the shadow, you can differentiate the equation x^2 + 15^2 = 5^2 with respect to time, and then solve for dx/dt. That will give you the rate at which the shadow is shrinking when the man is 5 meters from the building.
Good luck with the math, and remember to keep shining like the hilarious person you are! If you need any more jokes or help, feel free to ask!
To find the rate at which the man's shadow on the building is growing shorter, we can use similar triangles. The height of the man, the length of his shadow, and the distance from the man to the building form a similar triangle to the height of the building and the length of the shadow on the building.
Let's denote:
height of the man = h1 = 1.69 meters
rate of walking = v = 1.5 meters per second
distance of the light from the building = d = 15 meters
distance of the man from the building = x (which is 5 meters in this case)
Using similar triangles, we can establish the following ratio:
(h1 + h2) / h2 = (d + x) / x
where h2 is the height of the building and h1 + h2 is the total height.
Simplifying the equation, we get:
(1.69 + h2) / h2 = (15 + x) / x
Next, we differentiate the equation with respect to time:
[(1.69 + h2) / h2]' = [(15 + x) / x]'
Now, let's assume that h2 is constant over time and x changes with time. Also, let's denote the rate at which the man changes his distance from the building (dx/dt) as v1.
[(1.69 + h2) / h2]' = [(15 + x) / x]'
Differentiating both sides with respect to time:
[(1.69 + h2) / h2]' = [(15 + x) / x]'= [(15 + x)'] / x - [(15 + x) * (x') / x^2]
Since the rate of walking (v) is the derivative of x with respect to time, we can write:
v1 = v
Substituting this into the derivative equations, we get:
[(1.69 + h2) / h2]' = [(15 + x) / x]'= [(15 + x)'] / x - [(15 + x) * (v) / x^2]
Next, let's substitute the given values:
h1 = 1.69 meters
v = 1.5 meters per second
d = 15 meters
x = 5 meters
[(1.69 + h2) / h2]' = [(15 + 5) / 5]'= [(20)'] / 5 - [(20) * (1.5) / 5^2]
Simplifying this, we get:
[(1.69 + h2) / h2]' = [4]' - [20 * 1.5 / 25]
Since h1 is constant, we can consider it as 0 and simplify further:
[(1.69 + h2) / h2]' = [4]' - [0.6]
Now, the rate at which the man's shadow on the building is growing shorter is equal to [(1.69 + h2) / h2]'. Thus:
[(1.69 + h2) / h2]' = [4]'
Since the rate of walking is the derivative of x with respect to time, we have:
dx/dt = v1 = v = 1.5 meters per second
Therefore, the rate at which the man's shadow on the building is growing shorter when he is 5 meters from the building is 4 meters per second.
To find the rate at which the man's shadow on the building is shrinking, we need to use similar triangles. Let's set up a diagram to better understand the situation.
We have a right triangle formed by the man, his shadow, and the building. The height of the man represents one leg of the triangle, the length of his shadow represents the other leg, and the distance between the man and the building represents the hypotenuse of the triangle. Let's label the height of the man as h, the length of his shadow as s, and the distance between the man and the building as d.
Given that the man is 1.69 meters tall and his shadow is 15 meters away from the building, we can use similar triangles to set up the following ratio:
h/d = s/15
Now, let's differentiate this equation with respect to time since we're looking for the rate of change in the shadow's length:
(dh/dt)/d = (ds/dt)/15
We need to find the value of (ds/dt) when the man is 5 meters away from the building, which means d = 5. Now, let's substitute the given values into the equation and solve for (ds/dt):
(ds/dt)/15 = (dh/dt)/5
Since the man's height is not changing, dh/dt = 0. Thus, (ds/dt)/15 = 0, and solving for (ds/dt) gives us:
(ds/dt) = 0 * 15
Therefore, the rate at which the man's shadow on the building is shrinking when he is 5 meters from the building is 0 meters per second.