# AP Calculus question

<p>Hello, people! This is treasureyk again.
Here it goes:</p>

<p>A light shines from a pole 50ft high. A bll is dropped from the same height from a point 30 ft away from the light. How fast is the shadow of the ball moving along the ground 1/2 second later? (ball falls at 16t^2 ft in t seconds)</p>

<p>I'm a bit rusty on calculus, so take my solution with a grain of salt.</p>

<p>s(t)=16t^2 is the position function.</p>

<p>Take the derivative, v(t)=s'(t), to obtain velocity.</p>

<p>v(t)=s'(t)=32t</p>

<p>Speed is the absolute value of velocity.</p>

<p>v(1/2)=32(1/2)=16.</p>

<p>16 ft/s.</p>

<p>No, you need to do much more work than that; this is a related rates problem.</p>

<p>Basically you can set up a right triangle to get your equation relating your movement of the ball to the movement of the shadow. Then you will need to use your basic differentiation and some pythagorean to get yoru values of velocities and distances to solve your differantial equation.</p>

<p>Yeah, it's a differential equation, yuck.</p>

<p>haha, everyone shies away</p>

<p>First you need a diagram. Here are the instructions.
Draw a rectangle. Label the height 50ft and the width 30ft. The two vertical sides are the pole and ball path. Pick a point on the side representing the path of the ball. This will represent the ball at a general moment. From this point draw a dashed line to the rectangle's base and label it's length "h." Now, draw a straight line from the top of the pole through the ball and to the extended base of the rectangle. Where this line ends is the position of the shadow. Label the length of the line from that point to the bottom of the ball path as "x."</p>

<p>Now, the calculations must be done. Using the proportionality of the triangles in the picture you find the following:
x/h=(30+x)/50
Solve this and you find x=30h/(50-h). Now take the derivative with respect to time of each side (you need to think about this step, don't just copy me). After you differentiate you find x'=50h'/(50-h)^2. Now just plug in the numbers.</p>

<p>Hope this helps.</p>