A light shines frm the top of a pole 50ft high.A ball is dropped frm the same height frm a point 30ft away frm the light.How fast is the shadow of the ball moving along the ground 1/2sec later?(Assume the ball falls s=16t^2ft in t sec)
A light shines frm the top of a pole 50ft high.A ball is dropped frm the same height frm a point 30ft away frm the light.How fast is the shadow of the ball moving along the ground 1/2sec later?(Assume the ball falls s=16t^2ft in t sec)
hello,
I've attached a drawing of the situation.
You are dealing with 2 similar triangles. You can set up the proportion:
$\displaystyle \frac s{30} = \frac{30-s}y$. Solve for y because that's the length which the shadow of the ball is away from the pole:
$\displaystyle y=\frac{900}s - 30$ . You are told that $\displaystyle s = 16t^2$ . Substitute the variable s and you'll get the equation:
$\displaystyle y(t) = \frac{900}{16t^2}-30$
You know that speed is the first derivative wrt t of the length:
$\displaystyle y'(t) = -\frac{900}{8t^3}$
The speed at $\displaystyle t = \frac12$ is:
$\displaystyle \rm{speed} = y'\left(\frac12 \right) = -\frac{900}{8 \cdot \frac18} = -900$