A light is hung 15 ft above a straight horizontal path. If a man 6 ft tall is walking away from the light at the rate of 5 ft/sec, how fast is his shadow lengthening and at what rate is the tip of the man’s shadow moving?
depends on where the man is.
If the man's distance is x, and the shadow has length s, then
(x+s)/15 = s/6
Now just differentiate and plug in the distance when you finally deign to include it.
If the man is 8 feet away from the light?
To solve this problem, we can use similar triangles and related rates.
Let's consider two similar triangles - the triangle formed by the man's height, his shadow, and the ground, and the triangle formed by the light's height, the man's shadow, and the ground.
The height of the man's shadow will be the same as the height of the man himself, which is 6 ft.
Now, let's consider the horizontal distance between the man and the light. Since the man is walking away from the light, this distance is changing. Let's call it x.
We can set up a proportion using the similar triangles:
(Height of man's shadow) / (Height of man) = (Length of man's shadow) / (Distance between man and light)
6 / 6 = x / (x + 15)
Simplifying the proportion:
1 = x / (x + 15)
Cross multiplying:
x = x + 15
Subtracting x from both sides:
0 = 15
This implies that x = 0. However, this doesn't make sense in the context of the problem, as the man is walking away from the light. Hence, there is no solution to this problem.
Therefore, the man's shadow is not lengthening, and the tip of the man's shadow is not moving.
To solve this problem, we can use similar triangles and related rates.
Let's denote the length of the man's shadow as 'x' and the distance between the man and the light as 'y'. We need to find dx/dt (rate at which the shadow lengthens) and dy/dt (rate at which the tip of the man's shadow moves).
First, let's set up the proportion between the heights of the man and the length of his shadow:
6 ft / x = 15 ft / y
Cross-multiplying:
6y = 15x
Differentiating implicitly with respect to time 't':
6(dy/dt) = 15(dx/dt)
Simplifying:
(dy/dt) = 15/6 * (dx/dt)
(dy/dt) = 5/2 * (dx/dt) ---(1)
Now, let's establish the relationship between the distances x and y using the Pythagorean theorem:
x^2 + y^2 = (15 + 6)^2
x^2 + y^2 = 21^2
x^2 + y^2 = 441
Differentiating implicitly with respect to time 't':
2x(dx/dt) + 2y(dy/dt) = 0
Simplifying:
x(dx/dt) + y(dy/dt) = 0
Substituting the value of (dy/dt) from equation (1):
x(dx/dt) + y * (5/2 * (dx/dt)) = 0
Simplifying further:
x(dx/dt) + (5/2)y(dx/dt) = 0
(dx/dt)(x + (5/2)y) = 0
Since we know that (dx/dt) is non-zero (as the shadow is lengthening), we have:
x + (5/2)y = 0
Substituting the value of y from the proportion equation:
x + (5/2)(15x/6) = 0
Simplifying:
x + (25/4)x = 0
x(1 + 25/4) = 0
x(29/4) = 0
Therefore, x must be zero.
This means that as the man moves away from the light, his shadow will eventually disappear. Hence, the rate at which the man's shadow lengthens (dx/dt) is zero, and the rate at which the tip of the man's shadow moves (dy/dt) is also zero.
Thus, his shadow is not lengthening, and the tip of his shadow is not moving.