A man is 6ft tall and is walking at night straight toward a lighted street lamp at a rate of 5 ft/sec. If the lamp is 20 ft above the ground, find the rate at which the length of his shadow is changing.
If the man is x away from the lamp, and his shadow is s,
s/6 = (x+s)/20
s = 3/7 x
so,
ds/dt = 3/7 * dx/dt = 3/7 (-5) = -15/7 ft/s
I don't know if this is right because if you are using x for the distance the man is away from the light post then you cant set x equal to the rate at which the man is walking.
You are correct. However, dx/dt is the rate he is walking away. That is the value I used.
To find the rate at which the length of the man's shadow is changing, we can use similar triangles. The height of the man, the height of the lamp, the length of the man's shadow, and the distance between the man and the lamp form a set of similar triangles.
Let's denote the length of the man's shadow as "s" and the distance between the man and the lamp as "x". We need to find ds/dt, the rate at which the length of the shadow is changing, with respect to time.
Since the triangles are similar, we can set up the following proportion:
(man's height) / (length of shadow) = (lamp's height) / (distance from man to lamp)
Or, using the given values:
6ft / s = 20ft / x
To find ds/dt, we need to differentiate this equation with respect to time "t". But before we do that, we need to eliminate "x" from the equation. We know that the man is walking towards the lamp at a rate of 5 ft/sec, so the rate at which x is changing is -5 ft/sec (negative because he is getting closer to the lamp).
Differentiating both sides of the equation with respect to "t", we get:
(d/dt)(6ft / s) = (d/dt)(20ft / x)
To find the left-hand side of the equation, we need to differentiate 6ft/s with respect to "t". We can use the quotient rule:
(d/dt)(6ft / s) = (s * 0 - 6ft * ds/dt) / s^2 = -6ft * ds/dt / s^2
For the right-hand side, we differentiate 20ft/x with respect to "t". Since x is changing at a rate of -5 ft/sec, we have:
(d/dt)(20ft / x) = (x * 0 - 20ft * (-5ft/sec)) / x^2 = 100ft / x^2
Now we can rewrite our equation as:
-6ft * ds/dt / s^2 = 100ft / x^2
To solve for ds/dt, we can rearrange the equation:
ds/dt = (-6ft * s^2) / (100ft * x^2)
We know that s = 6ft and x = 20ft since these values are given in the question. Substituting these values into the equation, we get:
ds/dt = (-6ft * (6ft)^2) / (100ft * (20ft)^2)
Simplifying this expression, we find:
ds/dt = -0.032 ft/sec
Therefore, the rate at which the length of the man's shadow is changing is -0.032 ft/sec.