Applications of Derivatives Word Problems and Rate of Change

A street light is at the top of a 14 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 7 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 40 ft from the base of the pole?

Note: You should draw a picture of a right triangle with the vertical side representing the pole, and the other end of the hypotenuse representing the tip of the woman's shadow. Where does the woman fit into this picture? Label her position as a variable, and label the tip of her shadow as another variable. You might like to use similar triangles to find a relationship between these two variables.

Centerville is the headquarters of Greedy Cablevision Inc. The cable company is about to expand service to two nearby towns, Springfield and Shelbyville. There needs to be cable connecting Centerville to both towns. The idea is to save on the cost of cable by arranging the cable in a Y-shaped configuation. Centerville is located at in the -plane, Springfield is at , and Shelbyville is at . The cable runs from Centerville to some point on the -axis where it splits into two branches going to Springfield and Shelbyville. Find the location that will minimize the amount of cable between the 3 towns and compute the amount of cable needed. Justify your answer.
To solve this problem we need to minimize the following function of :

We find that has a critical number at
To verify that has a minimum at this critical number we compute the second derivative and find that its value at the critical number is , a positive number.
Thus the minimum length of cable needed is

Please see the attached file for the fully formatted problems.

Use Newton's method to approximate a root of the equation as follows.
Let be the initial approximation.
The second approximation is
and the third approximation is