A bright light on the ground illuminates a wall 12 meters away. A man walks from the light straight toward the building at a speed of 1.1 m/s. The man is 2 meters tall. When the man is 4 meters from the building, how fast is the length of his shadow on the building decreasing? (Give your answer correct to two decimal places.)
To find how fast the length of the man's shadow on the building is decreasing, we need to use related rates.
Let's assign variables to the given quantities:
- Let x be the distance between the man and the light source.
- Let y be the length of the man's shadow on the building.
We know:
- dx/dt = -1.1 m/s (The man is walking towards the building, so his distance from the light is decreasing.)
- x = 12 m (The man is initially 12 meters away from the light source.)
- h = 2 m (The height of the man)
First, let's find an equation relating x, y, and h using similar triangles:
x/y = (x+h)/h
Expanding this equation, we get:
xh = y(x+h)
Differentiating both sides with respect to time (t), we get:
(xh)' = (y(x+h))'
Using the product rule and chain rule, we have:
hx' + xh' = y'(x+h) + y(x+h)'
Since we want to find the value of y', the rate at which the length of the shadow is decreasing, let's solve for y':
y' = (hx' + xh' - y(x+h)') / (x+h)
Now, let's plug in the given values:
x = 12 m
h = 2 m
x' = -1.1 m/s
We need to find y' when the man is 4 meters from the building, so let's plug that in as well:
x = 4 m
Now we can substitute these values into the equation to find y':
y' = (hx' + xh' - y(x+h)') / (x+h)
y' = (2*(-1.1) + 12(-1.1) - y(4+2)') / (4+2)
Simplifying further:
y' = (-2.2 - 13.2 - 6y') / 6
6y' + 6y' = -2.2 - 13.2
12y' = -15.4
Dividing by 12:
y' = -15.4/12
y' ≈ -1.28
Therefore, the length of the man's shadow on the building is decreasing at a rate of approximately -1.28 m/s.
To find the rate at which the length of the man's shadow on the building is decreasing, we need to apply related rates.
Let's denote the length of the man's shadow on the building as "x" and the distance between the man and the light source as "d." We are given that the distance between the light source and the wall is 12 meters, so d = 12.
We know that the man is walking towards the building at a speed of 1.1 m/s, so the rate at which the distance between the man and the wall is changing can be represented as dx/dt = -1.1. The negative sign indicates that the man is getting closer to the wall.
Now, we need to find the relationship between the length of the man's shadow and the distance between the man and the wall. Since the man and the light source form similar triangles with the wall, we can use the property of similar triangles to establish a relationship.
The ratio of the man's height to the length of his shadow is constant and equal to the ratio of the distance between the light source and the wall to the distance between the man and the wall:
2/x = 12/d
To make the equation easier to work with, we can rearrange it as:
x = 2d/12
Substituting d = 12 into the equation:
x = 2(12)/12
x = 2
Now, we have an equation relating x and d, which we can differentiate with respect to time (t) to find the rate at which the length of the shadow is changing:
dx/dt = (d/dt)(2d/12)
Since we know that dx/dt = -1.1 and d = 12, we can substitute these values into the equation:
-1.1 = (d/dt)(2(12)/12)
Simplifying the equation:
-1.1 = (d/dt)(2)
Now we can solve for d/dt, which represents the rate at which the length of the man's shadow on the building is decreasing:
d/dt = -1.1 / 2
d/dt = -0.55
Therefore, the rate at which the length of the man's shadow on the building is decreasing is -0.55 meters per second.