Shadow Length A man 6 feet tall walks at a rate of 3 ft per second away from a light that is 16 ft above the ground (see figure). When he is 11 ft from the base of the light find the following.
(a) The rate the tip of the shadow is moving.
(b) The rate the length of his shadow is changing.
To find the rate at which the tip of the shadow is moving, we can use similar triangles. Let's call the distance between the man and the light "x".
The height of the man's shadow can be represented by the following ratio:
height of shadow / height of man = length of shadow / distance from man to light
Using this proportion:
x / 6 = 16 / (11 + x)
Next, we can cross-multiply and solve for "x":
16x = 6 * (11 + x)
16x = 66 + 6x
10x = 66
x = 6.6 ft
So, when the man is 11 ft from the base of the light, the distance between the man and the light is 6.6 ft.
Now, to find the rate at which the tip of the shadow is moving, we need to differentiate the equation with respect to time.
Differentiating both sides of the equation above:
(d/dt)(x / 6) = (d/dt)(16 / (11 + x))
To find the left side, we need to apply the quotient rule:
[(d/dt)x * 6 - x * (d/dt)6] / 6^2 = (d/dt)(16 / (11 + x))
Simplifying further:
(dx/dt * 6 - x * 0) / 36 = (d/dt)(16 / (11 + x))
(dx/dt * 6) / 36 = -16(dx/dt) / (11 + x)^2
Now, let's substitute the values we know:
(dx/dt * 6) / 36 = -16(dx/dt) / (11 + 6.6)^2
Simplifying again:
(dx/dt * 6) / 36 = -16(dx/dt) / 295.84
Now, we can simplify further to find the rate at which the tip of the shadow is moving:
(dx/dt * 6 / 36) = -16(dx/dt) / 295.84
6 = -36(dx/dt) / 295.84
6 * 295.84 = -36(dx/dt)
Solving for dx/dt:
36(dx/dt) = -(6 * 295.84)
36(dx/dt) = -1775.04
dx/dt = -1775.04 / 36
dx/dt = -49.31 ft/s
Therefore, the rate at which the tip of the shadow is moving is approximately -49.31 ft/s (negative sign indicates that the tip of the shadow is moving away from the light).
To find the rate at which the length of the shadow is changing, we can relate it to the rate at which the tip of the shadow is moving using similar triangles.
The ratio of the length of the shadow to the distance between the man and the light is the same as the ratio of the height of the man to his distance from the light:
length of shadow / distance from man to light = height of man / height of shadow
Using the values we know:
length of shadow / 6.6 ft = 6 ft / 16 ft
Simplifying this equation:
length of shadow = (6.6 ft * 6 ft) / 16 ft
length of shadow = 2.475 ft
Now, we can differentiate both sides of the equation with respect to time:
(d/dt)(length of shadow) = (d/dt)(2.475 ft)
Since the length of the shadow does not change with time, the derivative on the left side is zero:
0 = (d/dt)(2.475 ft)
Therefore, the rate at which the length of the shadow is changing is 0 ft/s.