Linty Fresh
Junior Member
- Joined
- Sep 6, 2005
- Messages
- 58
Hi all!
I'm having a bit of trouble with this related rates problem.
A street light is mounted at the top of a 15'foot pole. A man six feet tall walks away from the pole with a speed of 5 ft/s along a straight path. How fast is the tip of his shadow moving when he is 40 ft from the pole?
OK, so I constructed the standard triangle with x being the distance of the tip of the man's shadow (x=40+s, "s" being the distance from the man to his shadow). "y" is the height of the streetlight, and "z" is the hypotenuse of the triangle (distance from the street light to the tip of the man's shadow).
First, I found the length of x. Using similar triangles:
6'/15' = s/(40+s) --> s=26.7 and x=66.7'. As y = 15', x^2 + y^2 = z^2, and z=68.4'
Deriving x^2 + y^2 = z^2, we get 2x(dx/dt) + 2y(dy/dt) = 2z(dz/dt)
As dy/dt=0 (The street light does not move), we're left with 2x(dx/dt) = 2z(dz/dt)
Now I look at this, and it seems to me that dz/dt represents the speed of the man's shadow. dx/dt=5 ft/sec. Thus, solve for dz/dt, and I get 4.9 ft/sec. The book gives me 25/3 ft/sec.
What am I doing wrong? Many thanks!
I'm having a bit of trouble with this related rates problem.
A street light is mounted at the top of a 15'foot pole. A man six feet tall walks away from the pole with a speed of 5 ft/s along a straight path. How fast is the tip of his shadow moving when he is 40 ft from the pole?
OK, so I constructed the standard triangle with x being the distance of the tip of the man's shadow (x=40+s, "s" being the distance from the man to his shadow). "y" is the height of the streetlight, and "z" is the hypotenuse of the triangle (distance from the street light to the tip of the man's shadow).
First, I found the length of x. Using similar triangles:
6'/15' = s/(40+s) --> s=26.7 and x=66.7'. As y = 15', x^2 + y^2 = z^2, and z=68.4'
Deriving x^2 + y^2 = z^2, we get 2x(dx/dt) + 2y(dy/dt) = 2z(dz/dt)
As dy/dt=0 (The street light does not move), we're left with 2x(dx/dt) = 2z(dz/dt)
Now I look at this, and it seems to me that dz/dt represents the speed of the man's shadow. dx/dt=5 ft/sec. Thus, solve for dz/dt, and I get 4.9 ft/sec. The book gives me 25/3 ft/sec.
What am I doing wrong? Many thanks!