Math problems for August 11

1. Find the point on the parabola that is closest to the point (0,-3). ( Hint: Remember the distance formula that says the distance between the point ( ) and the point ( ) is . You might want to minimize the square of the distance, but then you should explain why that is relevant).

2. Find the maximum area of a rectangle that can be circumscribed about a given rectangle with length L and width W.

3. A model rocket is fired vertically upward from rest. Its acceleration for the first three seconds is a(t) = 60 t at which time the fuel is exhausted and it becomes a freely "falling" body. After 17 seconds, the rocket's parachute opens, and the (downward) velocity slows linearly to ft/s in 5 seconds. The rocket then "floats" to the ground at that rate.

(a) Determine the position function s and the velocity function v for all times t. Sketch the graphs of s and v.

(b) At what time does the rocket reach its maximum height, and what is that height?

(c) At what time does the rocket land?

4. How many points of intersection do the curves and have? First, experiment by graphing these curves on the same screen for various values of a . Then try and prove what you have discovered. In Maple, is a^x and is log[a](x). The answer to the question definitely depends on the value of a .