a light is hung 15 feet above a straight horizontal path. If a man is 6 feet tall is walking away from the light at a rate of 5 feet/sec.
a) how fast is his shadow lengthening?
b) at what rate is the tip of the man's shadow moving?
a) The man's shadow lengthens as he moves away from the light. Since the light is directly above the man, we can consider the problem as a similar triangles situation.
Using similar triangles, we can establish the following proportion:
Height of the man / Length of his shadow = Height of the light / Length of the light's shadow
6 / x = 15 / (x + y)
Where x represents the length of the man's shadow and y represents the length of the light's shadow.
Now, let's differentiate both sides of the equation with respect to time (t):
(d/dt)(6 / x) = (d/dt)(15 / (x + y))
Since the man is walking away from the light, the rate of change of his height is zero.
0 = (d/dt)(15 / (x + y))
Now, let's solve for the rate at which the man's shadow is lengthening, which is the value of (dy/dt):
0 = (d/dt)(15 / (x + y))
0 = 15 * (d/dt)(1 / (x + y))
0 = 15 * (-1 / (x + y)^2) * (d/dt)(x + y)
Since (d/dt)(x + y) represents the rate at which the man's shadow length is changing, we can substitute this value:
0 = 15 * (-1 / (x + y)^2) * (d/dt)(x + y)
Simplifying the equation further:
0 = (-15 / (x + y)^2) * (d/dt)(x + y)
We know that (d/dt)(x + y) is equal to the rate at which the man is moving away from the light, which is 5 feet/sec.
0 = (-15 / (x + y)^2) * 5
0 = (-75 / (x + y)^2)
Since the rate of change cannot be negative, the rate at which the man's shadow lengthens is 0. Therefore, the man's shadow is not lengthening.
b) Since the man's shadow is not lengthening, the tip of the man's shadow is not moving at all. So, the rate at which the tip of the man's shadow is moving is 0 feet/sec.
To solve this problem, we can use similar triangles and apply the concepts of rates of change.
Let's label the light as point L, the man as point M, and the tip of the shadow as point T.
a) To find how fast the man's shadow is lengthening, we need to calculate the rate of change of the length of the shadow with respect to time.
Let's represent the length of the man's shadow by s, and the distance between the man and the light by x.
From similar triangles, we know that the ratio of the length of the man's shadow (s) to the distance from the man to the light (x) is equal to the ratio of the height of the man (6 feet) to the height of the light (15 feet):
s/x = 6/15
To find the rate of change of the length of the shadow (ds/dt), we need to differentiate both sides of this equation with respect to time (t):
ds/dt = (6/15) * dx/dt
Since we know that dx/dt (the man's rate of change of distance from the light) is 5 feet/sec, we can substitute this value into the equation:
ds/dt = (6/15) * 5
ds/dt = 2 feet/sec
Therefore, the shadow is lengthening at a rate of 2 feet/sec.
b) To find at what rate the tip of the man's shadow is moving, we need to calculate the rate of change of the distance between the man and the tip of his shadow (x) with respect to time.
Again, using similar triangles, we know that the ratio of the distance between the man and the tip of his shadow (x) to the length of the man's shadow (s) is equal to:
x/s = 15/6
Differentiating both sides of this equation with respect to time (t) gives us:
dx/dt = (15/6) * ds/dt
Substituting ds/dt = 2 feet/sec, we can calculate dx/dt:
dx/dt = (15/6) * 2
dx/dt = 5 feet/sec
Therefore, the tip of the man's shadow is moving at a rate of 5 feet/sec.
To find the rate at which the man's shadow is lengthening and the rate at which the tip of the shadow is moving, we can use similar triangles and the chain rule from calculus. Let's break down the problem step by step:
a) How fast is his shadow lengthening?
Let's define some variables:
- Let x be the distance between the man and his shadow.
- Let y be the length of the man's shadow.
We can see that we have two similar triangles. The big triangle consists of the man's height, his shadow length, and the distance from the man to his shadow. The small triangle is formed by the man's shadow, the light above, and the distance from the light to the man.
Since the triangles are similar, we can write the following ratio:
x/y = (x + 6)/(y + 15)
Now, we will differentiate both sides of the equation with respect to time (t) using the chain rule:
(dx/dt) / y = (x + 6) / (dt/dt(y + 15)).
Notice that (dx/dt) is the rate at which the man is moving away from his shadow, which is given as 5 feet/sec. We can substitute this value into the equation:
5 / y = (x + 6) / (dy/dt).
Simplifying the equation, we get:
(dy/dt) = y(x + 6) / 5.
Substituting the values of x and y from the given information:
x = 15 feet (distance from the man to the light)
y = 6 feet (length of the man)
dy/dt = (6 * (15 + 6)) / 5
dy/dt ≈ 25.2 feet/sec.
Therefore, the rate at which the man's shadow is lengthening is approximately 25.2 feet/sec.
b) At what rate is the tip of the man's shadow moving?
To find the rate at which the tip of the man's shadow is moving, we need to differentiate the distance between the light and the man's shadow (x) with respect to time. The rate can be found by differentiating both sides of the equation using the chain rule:
(dx/dt) / y = (x + 6) / (dt/dt(y + 15)).
Substituting the given values:
5 / 6 = (15 + 6) / (dx/dt).
Simplifying the equation:
(dx/dt) = (21 * 6) / 5
(dx/dt) = 25.2 feet/sec.
Therefore, the tip of the man's shadow is moving at approximately 25.2 feet/sec.
Let
x = distance from light
s = length of shadow
Then
(x+s)/15 = s/6
s = 2/3 x
(a) ds/dt = 2/3 dx/dt = 2/3 * 5 = 10/3 ft/s
(b) 5 + 10/3 = 25/3 ft/s