A 6ft tall woman is walking away from a srteet lamp at 4 ft/sec that is 24ft tall. How fast is the length of her shadow changing?
Make a sketch. Draw a line from the top of the street lamp to the end of her shadow
I see two similar right-angled triangles.
Let the length of her shadow be x ft
let her distance to the street lamp be y
then 24/(x+y) = 6/x
which simplifies to
3x = y
3dx/dt = dy/dt
dx/dt = 4/3 ft/s
In this type of question, the length of her shadow is growing at a uniform rate, independent of the time passed or her position.
Secondly, had the question been, "how fast is her shadow moving?" you would have to add her speed to the 4/3.
It depends upon her distance from the swtreet lamp at the time. Did they not tell you that distance?
no time given
To find out how fast the length of the woman's shadow is changing, we can use similar triangles and apply the chain rule from calculus. Let's break down the problem and set up the solution step by step:
1. Identify the variables:
- Let h be the height of the woman (6 ft).
- Let x be the distance between the woman and the street lamp.
- Let L be the length of the shadow.
- Let t be the time (in seconds).
2. Determine the given rates:
- The woman is walking away from the street lamp at a rate of 4 ft/sec.
- The height of the street lamp is 24 ft.
3. Establish the relationship between the variables:
- The top of the woman's shadow, the top of the woman, and the top of the street lamp form similar triangles.
- Therefore, we know that (L + h)/L = x/24.
4. Differentiate both sides of the equation with respect to time (t):
- Differentiating gives (dL/dt + dh/dt) / L = dx/dt / 24.
5. Solve for the rate of change of the shadow length (dL/dt):
- Rearranging the equation, we get dL/dt = (dx/dt / 24) * L - (dh/dt / L) * (L + h).
6. Substitute the known values:
- dx/dt = -4 ft/sec (negative because the woman is moving away from the street lamp).
- L = x/4 (since the triangles are similar, the height of the woman to the height of the shadow follows the same ratio).
- dh/dt = 0 ft/sec (the woman's height does not change).
7. Plug in the values and calculate dL/dt:
- dL/dt = (-4 / 24) * (x / 4) - (0 / (x/4)) * (x/4 + 6).
8. Simplify the equation:
- dL/dt = (-1/6) * (x/4) + 0.
- dL/dt = -x/24.
Therefore, the rate at which the length of the woman's shadow is changing is given by dL/dt = -x/24 ft/sec.