Final answer:
By using the principles of related rates and similar triangles, the rate at which a woman's shadow is getting taller as she walks towards a projector can be calculated when she is 10.0 m away from it.
Step-by-step explanation:
To find out how fast the woman's shadow is getting taller when she is 10.0 m from the projector, we need to use similar triangles and the concept of related rates in calculus. Let's denote the height of the projector as h_p (1.2 m), the height of the woman as h_w (1.75 m), the distance between the projector and the screen as l (50 m), the distance of the woman from the projector as x, and the length of the shadow on the screen as y.
Using similar triangles, the ratio of the height of the object to the length of the shadow should equal the ratio of the height of the projector to the total distance from the screen to the projector:
(h_w / y) = (h_p / (l - x))
Since the woman is walking towards the projector at 1.0 m/s, and we want to find the rate at which the shadow's length is changing when the woman is 10.0 m away, we set
x = 10.0 m
and differentiate both sides of the equation with respect to time t:
d(h_w) / dt = d(h_p) / dt * ((l - x) / y) + h_p * d(l - x) / dt / (y^2) * (-dy / dt)
Since both h_p and h_w are constant, their derivatives with respect to time are zero. We are left with:
0 = h_p * (-(dx/dt) / (y^2)) * (-dy/dt)
We know that dx/dt is -1.0 m/s (since the woman is walking towards the projector, x decreases with time). Re-arranging the terms and solving for dy/dt will yield the rate at which the shadow's height is changing:
dy/dt = (h_p / y^2) * dx/dt
To find the value of y when x = 10.0 m, we use the initial similar triangles equation and solve for y:
y = (h_w * (l - x)) / h_p
Now, we substitute y with the derived expression and put in the known values to find the rate dy/dt when x = 10.0 m. The calculation will give us the answer to how quickly the shadow is getting taller.