Final answer:
To find the rate at which the tip of the shadow is moving and the rate at which the length of the shadow is changing when the person is 10 feet from the base of the light, we use similar triangles and the concept of the derivative. The rate at which the tip of the shadow is moving can be found by differentiating the equation for the length of the shadow with respect to time. The rate at which the length of the shadow is changing can also be found by differentiating the equation for the length of the shadow with respect to time.
Step-by-step explanation:
To answer both parts (a) and (b), we need to use similar triangles and the concept of the derivative. Let's call the distance of the person from the base of the light 'x' and the length of the shadow 'y'.
(a) To find the rate at which the tip of the shadow is moving, we need to differentiate the equation for the length of the shadow with respect to time. Assuming the person is moving away from the light source, we have:
dy/dt = (dx/dt) * (dy/dx)
Since the person is moving 10 feet from the base of the light, we can substitute x = 10 into the equation. The rate at which the tip of the shadow is moving is given by:
dy/dt when x = 10
(b) To find the rate at which the length of the shadow is changing, we need to differentiate the equation for the length of the shadow with respect to time:
dy/dt = (dx/dt) * (dy/dx)
We can substitute x = 10 and solve for dy/dt to find the rate at which the length of the shadow is changing when the person is 10 feet from the base of the light.