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 2.2 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?
What I did was
x/12=x+y/2
2x=12x+12y
10x=10y , took derivative
-10/12 dx/dy(2.2(
=-2.64 , I did not know what to do with the part when it says , when the man is 4 meters a way
Since you did not define what your x and y are, I have no way of checking what you did
In my sketch, I let the distance from the man to the light be x m, and the length of his shadow as y m
then by ratios ,
2/x = y/12
xy = 24
x dy/dt + y dx/dt = 0
when x=4, y=6 and dx/dt = 2.2
4dy/dt + 6(2.2) = 0
dy/dt = -13.2/4 = -3.3 m/s
the negative implies that his shadow is shrinking as expected when he walks towards the wall
To find out how fast the length of the man's shadow on the building is decreasing when he is 4 meters away from the building, we can use related rates.
Let's define some variables:
x = the distance of the man from the light source (on the ground)
y = the length of the man's shadow on the building
We know that the man is walking straight towards the building, so the distance he is from the light source (x) is decreasing. We are given that dx/dt = -2.2 m/s (negative because x is decreasing).
We are also asked to find how fast the length of the shadow is decreasing, which is dy/dt.
Now, let's set up an equation relating x and y. We can use similar triangles to relate the distances on the ground and the wall. The ratio of the corresponding sides will remain constant.
x/12 = (x + y)/2
Next, let's differentiate this equation with respect to time (t) to get the rates of change:
(d/dt)(x/12) = (d/dt)((x + y)/2)
To find dx/dt, we can differentiate x/12 with respect to t:
(dx/dt)/12 = (dx/dt + dy/dt)/2
Now, let's substitute the given values:
dx/dt = -2.2 m/s
-2.2/12 = (-2.2 + dy/dt)/2
Simplifying the equation:
-0.1833 = -1.1 + dy/dt/2
dy/dt/2 = -0.1833 + 1.1
dy/dt/2 = 0.9167
Multiply both sides by 2 to isolate dy/dt:
dy/dt = 2 * 0.9167
dy/dt = 1.8333 m/s
So, when the man is 4 meters away from the building, the length of his shadow on the building is decreasing at a rate of 1.8333 m/s.