A spotlight on the ground is shining on a wall 12m away. If a woman 2m tall walks from the spotlight toward the building at a speed of 0.6m/s, how fast is the length of her shadow on the building decreasing when she is 2m from the building?
Nevermind, got it.
To find the rate at which the length of the woman's shadow on the building is decreasing, we need to use related rates. We can start by drawing a diagram to visualize the situation.
Let's label the following:
- Distance from the spotlight to the woman as x (which is decreasing)
- Distance from the woman to the building as y (which is also decreasing)
- The length of the shadow on the building as s (which is the quantity we want to find)
From the given information, we have:
- x = 12m (constant)
- The woman's height is 2m
- The rate at which the woman is approaching the building is dx/dt = -0.6m/s (negative because the distance is decreasing)
Now, let's establish some relationships between the variables:
- Using similar triangles, we know that s/x = (s + 2)/y (since the height of the woman's shadow on the building is s + 2)
- We can rearrange the equation to solve for y: y = (x * (s + 2)) / s
To find how fast the length of the shadow is decreasing, we need to find ds/dt (the rate of change of s).
First, differentiate the equation with respect to time (t):
dy/dt = (dx/dt * (s + 2) + x * ds/dt) / s
Since dy/dt is not given and dx/dt is given (-0.6m/s), we can rearrange the equation to solve for ds/dt:
ds/dt = (s * dy/dt - dx/dt * (s + 2)) / x
Now we can substitute the known values into the equation:
- s = (2m) (the woman's initial height before she approaches the building)
- x = 12m (distance from the spotlight to the woman, constant)
- dx/dt = -0.6m/s (rate at which the woman is approaching the building)
- dy/dt = ? (the rate at which y is changing)
Since we are asked to find the rate at which the length of the shadow on the building is decreasing when the woman is 2m from the building (y = 2m), we need to find dy/dt at that moment.
To find dy/dt when y = 2m, we can use the relationship established earlier:
y = (x * (s + 2)) / s
Substituting the known values, we get:
2 = (12 * (s + 2)) / s
Solving for s, we have:
2s = 12(s + 2)
2s = 12s + 24
10s = 24
s = 2.4m
Now we have the value of s (the length of the shadow on the building when the woman is 2m from the building, which is 2.4m).
Substituting the values into the equation ds/dt = (s * dy/dt - dx/dt * (s + 2)) / x:
ds/dt = (2.4 * dy/dt - (-0.6) * (2.4 + 2)) / 12
ds/dt = (2.4 * dy/dt + 3.6) / 12
ds/dt = (0.2 * dy/dt + 0.3)
We don't know the exact value of dy/dt without further information. However, this is the equation that represents the relationship between the changing length of the shadow (ds/dt) and the changing distance from the spotlight to the woman (dy/dt) at the given moment.