(b) Find the dimensions of the triangle and the square that produce a maximum area. Justify your answer. (Area of an equilateral triangle = 2 3 , 4 x where x = side of triangle)
Answers to Worksheet 2 on Optimization Note: Only the answers are shown below. Students must also justify their answer. See the article on AP Central called “On the Role of Sign Charts in the AP Calculus Exam for Justifying Local or Absolute Extrema” found at to see how justifications for absolute maximums and minimums should be written. 1. The maximum size of the population of bacteria is 387.298 (so 387 bacteria) and it occurs when t = 7.750 weeks. 2. The minimum velocity is 7, and it occurs when t = 3. The maximum velocity is 15.333, and it occurs when 4 . 3 t 3. The minimum cost is $330, and it occurs when the dimensions of the tank are 4 m x 3 m x 3 m. 4. The maximum area is 0.281, and it occurs when x = 0.860. The minimum area is 0.121, and it occurs when x = 0.25. 5. The minimum time is 1.733 hr when x = 1.5 mi. so he should land the boat 4.5 mi. from the house. 6. The minimum cost is $996,862.70 when the point P is 6.803 mi. from B . 7. The minimum area occurs when the side of the triangle is 1.883 and the side of the square is 1.087. The maximum area is occurs when the side of the triangle is 0 and the side of the square is 3.333.
P CALCULUS AB WORKSHEET ON OPTIMIZATION AND PARTICLE MOTION Work the following on notebook paper. Use your calculator on 2 - 6, and give decimal answers correct to three decimal places. 1. A rectangle is bounded by the x -axis and the parabola 2 9 y x . What length and width should the rectangle have to that its area is a maximum? ___________________________________________________________________________________________ 2. A cylindrical container has a volume of 3 150cm . Find the radius and height of the cylinder so that its surface area will be a minimum. Use Calculus to find and to justify your answer. (Volume of a cylinder = 2 r h . Surface area of a cylinder = 2 2 2 rh r . ___________________________________________________________________________________________ 3. A drilling rig 12 miles offshore is to be connected to a refinery on shore, 20 miles down the coast from Rig the rig by using underwater pipe from the rig to point P and land-based pipe from point P to the refinery. If underwater pipe costs $5000 per mile and land-based 12 pipe costs $3500 per mile, how far should point P be from the refinery to minimize the cost? What will the cost be? Use Calculus to find and justify your answer. Refinery
4. (Modified from 2010) A zoo sponsored a one-day contest to name a new baby elephant. Zoo visitors deposited entries into a box between noon ( t = 0) and 8 PM. At 8 PM, volunteers began to process the entries at a rate modeled by the function 2 3 30 298 976 P t t t t , where P t gives the number of hundreds of entries per hour for 8 12 t . At what time t , 8 12 t , were the entries being processed most quickly? Use Calculus to find and justify your answer. 5. The velocity of a particle at time t is given by 2 3 ln 2 for 0 6 v t t t . Find the acceleration of the particle each time its velocity is zero.
- Fall '19
- Derivative, triangle, particle motion