To find out how fast the ball's shadow is moving along the ground, we need to calculate the rate of change of the shadow's position with respect to time. In other words, we need to find the derivative of the position of the shadow with respect to time.
Let's start by setting up a coordinate system. Let the top of the pole be the origin (0, 0), and let the positive x-axis point to the right along the ground. The light source is located at (0, 50), and the ball is dropped from a point 30 ft away from the light, which means its initial position can be represented as (30, 50).
The position of the ball can be expressed as (x(t), y(t)), where x(t) denotes the horizontal position of the ball and y(t) denotes its vertical position at time t.
We are given the equation for the vertical position of the ball as y(t) = 50 - 16t^2.
To find the horizontal position of the ball, x(t), let's consider the following: the shadow of the ball will always line up with the point directly beneath the ball. Since the ball was dropped from a point 30 ft away from the light source, the x-coordinate of the ball's position is equal to 30.
Therefore, x(t) = 30 for any value of t.
Now, let's find the derivative of x(t) with respect to t:
dx(t)/dt = d(30)/dt = 0
Since the shadow's horizontal position does not change over time, the derivative is zero, indicating that the shadow is not moving horizontally along the ground.
To find the rate of change of the shadow's vertical position with respect to time, we need to find dy(t)/dt.
dy(t)/dt = d(50 - 16t^2)/dt
= -32t
Now, let's evaluate dy(t)/dt at t = 1/2 sec:
dy(1/2)/dt = -32(1/2)
= -16
Therefore, the shadow of the ball is moving downward along the ground at a rate of 16 ft/s 1/2 second later.
In summary, the ball's shadow is moving downward at a speed of 16 ft/s along the ground 1/2 second later.