A light is on the ground 20m from a barn. A 2m tall llama walks from the light directly toward the side of the barn at 1m/s. How fast is the height of the llama's shadow on the barn changing when he's 14m from the barn? (draw and label a nice diagram first!)
If x is the diatsnace from the light to the llama, and h is the height of the shadow on the wall, then using similar triangles,
h/20 = 2/x
hx = 40
At 14m from the barn, x=6 and h = 40/6 = 20/3
hx' + xh' = 0
(20/3)(1) + 6h' = 0
h' = -10/9
Why did the llama cross the road? Because it wanted to measure the height of its shadow on the barn! Here's a delightful diagram to help us visualize the situation:
Light
* ♜ Llama
|\
| \
| \
| \
| \
| \
| \
--------------|--------------
Barn Shadow
Now, let's solve this puzzle! We want to find how fast the height of the llama's shadow on the barn is changing, given that the llama is 2m tall and moving towards the barn at a speed of 1m/s.
At any given point, we can create a right triangle between the llama, the light source, and the top of the barn. Let's call the distance from the llama to the barn x, and the height of the llama's shadow on the barn h.
Using similar triangles, we can see that the ratio of the llama's height to the length of the shadow is always constant. Therefore, we have:
h/2 = (x + 20)/x
Simplifying this equation, we get:
h = (2*(x + 20))/x
Now, we can differentiate both sides of the equation with respect to time (t). Remember, we want to find dh/dt (the rate of change of the height of the shadow):
dh/dt = [2(x + 20)' * x - 2(x + 20) * x'] / x^2
Since the llama is walking directly towards the barn, x is changing over time. We know that dx/dt = -1 m/s (negative sign indicates it's moving towards the barn). Substituting this information, we get:
dh/dt = [2(x + 20) * 1 - 2(x + 20) * (-1)] / x^2
Now, we can plug in the value x = 14m (as stated in the question) and calculate the rate of change of the height of the shadow. Or...
I could tell you a joke instead! What do you call a llama with no shadow? Invisible!
To solve the problem, let's start by drawing a diagram:
Llama
|-------------|
| | <- Barn
|-------------|
Light Source
Let's assign variables to the given information:
- Distance from the light to the barn = 20m (L)
- Height of the llama = 2m (H)
- Distance from the llama to the barn = 14m (x)
- Rate at which the llama is approaching the barn = 1m/s (dx/dt)
- Rate at which the height of the llama's shadow is changing on the barn = ? (dh/dt)
We need to find the rate at which the height of the llama's shadow on the barn is changing. To do this, we can use similar triangles.
Let's consider the larger right triangle formed by the llama, its shadow, and the distance from the llama to the barn. The height of the triangle will represent the height of the llama's shadow on the barn.
Using the similar triangles, we can set up the following equation:
H / (L + x) = h / x
Where:
- H is the height of the llama
- L + x is the total distance from the light to the barn
- h is the height of the llama's shadow on the barn
To find dh/dt, we need to differentiate both sides of the equation with respect to time (t):
d/dt (H / (L + x)) = d/dt (h / x)
Next, we can solve for dh/dt by substituting the given values into the equation.
To find the rate at which the height of the llama's shadow on the barn changes, we can use similar triangles and differentiation.
First, let's draw a diagram to visualize the situation:
---------------- ----------------
| BARN |
| | L
| | L
| LLAMA | L
| | L
-----------------------------------
In the diagram, the vertical line on the left represents the light source, the horizontal line represents the ground, and the rectangle on the right represents the barn. The height of the llama is labeled as "L."
We need to find how fast the height of the shadow is changing when the llama is 14m from the barn.
Let's label the following variables:
- Distance from the light to the barn = d (20m)
- Distance from the llama to the barn = x (unknown but given as 14m)
- Height of the llama = y (2m)
Since the triangles formed by the llama and its shadow are similar, we can set up the following proportion:
shadow height / barn height = (distance from the light to the barn - distance from the llama to the barn) / distance from the llama to the barn
Let's substitute the values we know:
shadow height / 2 = (20 - x) / x
To find the shadow height, we can rearrange the equation:
shadow height = (2 * (20 - x)) / x
Now, to find how fast the shadow height is changing, we differentiate the equation with respect to time (t):
d(shadow height) / dt = (d/dt)[(2 * (20 - x)) / x]
Applying the quotient rule:
d(shadow height) / dt = (2 * (x * 0) - (20 - x) * 1) / (x^2)
Simplifying:
d(shadow height) / dt = -(20 - x) / (x^2)
Now, let's substitute the value of x (14m) into the equation to find the rate of change of the shadow height when the llama is 14 meters from the barn:
d(shadow height) / dt = -(20 - 14) / (14^2)
Simplifying:
d(shadow height) / dt = -6 / 196
Therefore, the rate at which the height of the llama's shadow on the barn is changing when he's 14m from the barn is approximately -0.0306 m/s.